2 Preliminaries

The basic system that we rely on is the system $\mathcal {A}^\omega =\textsf {WE}\mbox {-}\textsf {PA}^\omega +\textsf {QF}\mbox {-}\textsf {AC}+\textsf {DC}$ for classical analysis in all finite types as commonly used in proof mining, formalized via (a weakly extensional variant of) Peano arithmetic in all finite types together with a few choice principles (see, e.g., [Reference Kohlenbach25] where this notation for the system was, presumably, first introduced). As all systems introduced here will be extensions of this system $\mathcal {A}^\omega $ , we in this section sketch the essential features relevant for this paper. For any further details, we refer to the works [Reference Kohlenbach26, Reference Troelstra60].

Here, we follow the definition of weakly extensional Peano arithmetic in all finite types $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ as, for example, given in [Reference Kohlenbach26] (see also [Reference Troelstra60]) and, in that way, we do not recall all the defining features $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ here and only focus on the four main aspects which are relevant in detail for this paper. In general, we denote function types using the bracket notation used in [Reference Kohlenbach26], that is, $\rho (\tau )$ is the type of functions that map objects of type $\tau $ to objects of type $\rho $ , and we use T to denote the set of all finite types as usual, that is,

$$\begin{align*}0\in T,\quad \rho,\tau\in T\rightarrow \rho(\tau)\in T. \end{align*}$$

As usual, we denote pure types by natural numbers by setting $n+1:=0(n)$ . The four central properties of $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ that we need here are that, for one, the only primitive relation is equality at type $0$ (denoted by $=_0$ ) and higher-type equality is only defined as an abbreviation via recursion with

$$\begin{align*}x^{\tau(\xi)}=_{\tau(\xi)}y^{\tau(\xi)}:=\forall z^{\xi}\left( xz=_\tau yz\right). \end{align*}$$

For another, $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ crucially does not contain the full extensionality principles

(E _ρ,τ) $$\begin{align} \forall x^{\tau(\rho)},y^{\rho},{y'}^{\rho}\left(y=_{\rho}y'\to xy=_{\tau}xy'\right). \end{align}$$

Instead, it only contains the quantifier-free extensionality rule

(QF–ER) $$\begin{align} \frac{A_0\to s=_\rho t}{A_0\to r[s/x^\rho]=_\tau r[t/x^\rho]} \end{align}$$

where $A_0$ is a quantifier-free formula, s and t are terms of type $\rho $ and r is a term of type $\tau $ . This lack of full extensionality is essential for establishing results on program extraction from classical proofs as the full extensionality axiom is not admissible in the context of the Dialectica interpretation, the main tool used later extract bounds from (classical) proofs. However, the extensionality rule is admissible in that context and so, from an applied perspective, serves the much-needed purpose of reintroducing an admissible fraction of extensionality back into the system to be practically as flexible as possible when dealing with proofs from the literature. We refer to [Reference Kohlenbach26] for a detailed discussion on this.

Further, $\textsf {WE}\text {-}\textsf {PA}^\omega $ contains constants ${\underline {R}}_{\underline {\rho }}$ for simultaneous primitive recursion in the sense of Gödel [Reference Gödel13] and Hilbert [Reference Hilbert18] as governed by the axioms

( R ) $$\begin{align} \begin{cases} (R_i)_{\underline{\rho}}0\underline{y}\underline{z}=_{\rho_i}y_i\\ (R_i)_{\underline{\rho}}(Sx)\underline{y}\underline{z}=_{\rho_i}z_i(\underline{R}_{\underline{\rho}}x\underline{y}\underline{z})x \end{cases}\text{ for }i=1,...,k \end{align}$$

where $\underline {\rho }=\rho _1,\dots ,\rho _k$ is a tuple of types, $\underline {y}=y_1,\dots ,y_k$ with $y_i$ of type $\rho _i$ and $\underline {z}=z_1,\dots ,z_k$ with $z_i$ of type $\rho _i(0)\underline {\rho }^t$ where we write $\underline {\rho }^t:=(\rho _k)\dots (\rho _1)$ .

Lastly, due to the inclusion of the combinators of Schönfinkel [Reference Schönfinkel54] in the language of $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ , the system allows the definition of $\lambda $ -abstraction in the sense that for any term t of type $\tau $ and any variable x of type $\rho $ , we can construct a term $\lambda x.t$ of type $\tau (\rho )$ such that the free variables of $\lambda x.t$ are exactly those of t without x and so that

$$\begin{align*}\textsf{WE}\mbox{-}\textsf{PA}^\omega\vdash (\lambda x.t)(s)=_\tau t[s/x] \end{align*}$$

for any term s of type $\rho $ .

Next to $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ , we define, as usual, the principle of quantifier-free choice $\textsf {QF}\mbox {-}\textsf {AC}$ , that is,Footnote ⁷

(QF–AC) $$\begin{align} \forall \underline{x}\exists\underline{y}A_0(\underline{x},\underline{y})\to \exists\underline{Y}\forall\underline{x}A_0(\underline{x},\underline{Y}\underline{x}) \end{align}$$

with $A_0$ quantifier-free and where the types of the variable tuples $\underline {x}$ , $\underline {y}$ are arbitrary, as well as the principle of dependent choice $\textsf {DC}$ defined as the collection of $\textsf {DC}^{\underline {\rho }}$ for all tuples of types $\underline {\rho }$ with

(DC ^ρ ) $$\begin{align} \forall x^0,\underline{y}^{\underline{\rho}}\exists \underline{z}^{\underline{\rho}} A(x,\underline{y},\underline{z})\to \exists \underline{f}^{\underline{\rho}(0)}\forall x^0 A(x,\underline{f}(x),\underline{f}(S(x))) \end{align}$$

where $\underline {f}^{\underline {\rho }(0)}$ stands for $f_1^{\rho _1(0)},\dots ,f_k^{\rho _k(0)}$ and A may now be arbitrary.

Over $\mathcal {A}^\omega $ , we will have to rely on some chosen representation of the real numbers as a Polish space and for that we follow definitions and conventions given in [Reference Kohlenbach26]. In particular, rational numbers are represented using pairs of natural numbers and, in that context, we fix the same pairing function j as in [Reference Kohlenbach25]:

$$\begin{align*}j(n^0,m^0):=\begin{cases}\min u\leq_0(n+m)^2+3n+m[2u=_0(n+m)^2+3n+m]&\text{if existent},\\0^0&\text{otherwise}.\end{cases} \end{align*}$$

The usual arithmetical operations $+_{\mathbb {Q}},\cdot _{\mathbb {Q}},\vert \cdot \vert _{\mathbb {Q}}$ , etc., are then primitive recursively definable through terms that operate on such codes and the usual relations $=_{\mathbb {Q}}$ , $<_{\mathbb {Q}}$ , etc., are definable via quantifier-free formulas.

For real numbers we then rely on a representation via fast converging Cauchy sequences of rational numbers with a fixed Cauchy modulus $2^{-n}$ (see [Reference Kohlenbach26] for details), that is, via objects of type $1$ , and we consider $\mathbb {N}$ and $\mathbb {Q}$ as being embedded in that representation via the constant sequences. Also here, the usual arithmetical operations like $+_{\mathbb {R}}$ , $\cdot _{\mathbb {R}}$ , $\vert \cdot \vert _{\mathbb {R}}$ , etc., are primitive recursively definable through closed terms and the relations $=_{\mathbb {R}}$ and $<_{\mathbb {R}}$ , etc., now operating on type $1$ objects, are representable by formulas of the underlying language. Naturally, these relations are not decidable anymore but are given by $\Pi ^0_1$ - and $\Sigma ^0_1$ -formulas, respectively.

In the context of this representation of reals, we will later rely on an operator $\widehat {\cdot }$ which allows for an implicit quantification over all such fast-converging Cauchy sequences of rationals. Following [Reference Kohlenbach26], we define this operator via

$$\begin{align*}\widehat{x}n:=\begin{cases}xn&\text{if }\forall k<_0n\left( \vert xk-_{\mathbb{Q}}x(k+1)\vert_{\mathbb{Q}}<_{\mathbb{Q}}2^{-k-1}\right),\\ xk&\text{for } k<_0n \text{ least with }\vert xk-_{\mathbb{Q}}x(k+1)\vert_{\mathbb{Q}}\geq_{\mathbb{Q}}2^{-k-1}\text{ otherwise},\end{cases} \end{align*}$$

turning x of type $1$ into a fast-converging Cauchy sequence $\widehat {x}$ , and we refer to [Reference Kohlenbach26] for any further discussions of its properties.

In the context of the bound extraction theorems later on, we will rely on a canonical selection of a Cauchy sequence representing a given real number. Naturally, such an association will be noneffective. However, it will suffice that the operation behaves well enough w.r.t. the notion of majorization. Following [Reference Kohlenbach25], this can be achieved for non-negative numbers via the function $(\cdot )_\circ $ defined by

$$\begin{align*}(r)_\circ(n):=j(2k_0,2^{n+1}-1), \end{align*}$$

where

$$\begin{align*}k_0:=\max k\left[\frac{k}{2^{n+1}}\leq r\right]. \end{align*}$$

Later, we will need an extension of this function $(\cdot )_\circ $ to all real numbers such that we retain these nice properties regarding majorizability and so, for $r<0$ , we consider $(r)_\circ $ to be defined byFootnote ⁸

where

$$\begin{align*}\bar{k}_0:=\max k\left[\frac{k}{2^{n+1}}\leq \vert r\vert\right]. \end{align*}$$

Then $(r)_\circ (n)=-_{\mathbb {Q}}(\vert r\vert )_\circ (n)$ and we get the following lemma containing exactly the properties that we later need for this notion to be useful in the context of majorizability (extending Lemma 2.10 from [Reference Kohlenbach25]):

Lastly, we write $r_\alpha $ for the unique real represented by $\widehat {\alpha }$ for a given sequence $\alpha \in \mathbb {N}^{\mathbb {N}}$ and we sometimes write $[\alpha ](n)$ for the rational number represented by the n-th element of that sequence for better readability.

In terms of notation, we want to note that, to enhance readability, we will omit the subscripts of the arithmetical operations for $\mathbb {R}$ everywhere. Further, we will omit the operation $\cdot _{\mathbb {R}}$ often altogether. Similarly, we will also omit types of variables whenever convenient. Also, we will almost always omit types or related subscripts in proofs. Lastly, we throughout denote the powerset of a set X by $2^X$ .

3 Systems for algebras of sets

In this section, we develop the underlying systems on which we will later bootstrap our treatment of probability contents and probability measures as well as of all the other respective extensions discussed before. As such, we begin with a treatment of algebras of sets as the most basic underlying algebraic notion that is essential for the theory of contents. For references on these basic definitions and their properties, if nothing else is mentioned otherwise, we mainly refer to [Reference Bhaskara Rao and Bhaskara Rao6].

The approach that we take towards a formal system for algebras of sets is to use abstract types to represent both the underlying ground set $\Omega $ as well as the algebra $S\subseteq 2^\Omega $ . One then has to restore the structure of S as a collection of subsets over $\Omega $ with certain operations on them by including additional constants that reintroduce these operations in this abstract setting.

Concretely, to form a system for the treatment of algebras, we extend the previously discussed set of types T by two new abstract types $\Omega $ and S, forming the extended set of types $T^{\Omega ,S}$ defined by

$$\begin{align*}0,\Omega,S\in T^{\Omega,S},\quad \rho,\tau\in T^{\Omega,S}\rightarrow \rho(\tau)\in T^{\Omega,S}, \end{align*}$$

and, over the resulting language, we then utilize this augmented set of types to introduce the following new constants to induce the usual structure on the set represented by S in relation to $\Omega $ as mentioned before:

• • $\mathrm {eq}$ of type $0(\Omega )(\Omega )$ ;

• • $\in $ of type $0(S)(\Omega )$ ;

• • $\cup $ of type $S(S)(S)$ ;

• • $(\cdot )^c$ of type $S(S)$ ;

• • $\emptyset $ of type S;

• • $c_\Omega $ of type $\Omega $ .

The constant $\mathrm {eq}$ serves as an abstract account of the equality relation between objects of type $\Omega $ while the constant $\in $ serves as an abstract account of the element relation between elements as objects of type $\Omega $ and sets as objects of type S. The constants $\cup $ and $(\cdot )^c$ reintroduce the respective operations of union and complement for the abstract type S and $\emptyset $ provides a constant representing the empty set. The constant $c_\Omega $ in particular is intended to witness that the underlying set $\Omega $ is nonempty. We often simply write $A^c$ instead of $(A)^c$ for A of type S. Further, we abbreviate $\in x A =_0 0$ by $x\in A$ and, similarly, we write $x\not \in A$ for $\in x A\neq _0 0$ . Lastly, we define $\Omega :=\emptyset ^c$ as a notation for the top element of S.Footnote ⁹ Also regarding notation, we introduce intersections as an abbreviation by defining

$$\begin{align*}A\cap B:=(A^c\cup B^c)^c \end{align*}$$

for terms $A^S, B^S$ .

We write $x=_\Omega y$ as an abbreviation of $\mathrm {eq} xy=_00$ for objects $x^\Omega ,y^\Omega $ . Using $\in $ , we introduce equality on S via the following abbreviation: for $A^S$ and $B^S$ , we define

$$\begin{align*}A =_S B:\equiv \forall x^\Omega(x\in A \leftrightarrow x \in B). \end{align*}$$

Note that $=_S$ clearly is, provably, an equivalence relation. Furthermore, we introduce the abbreviation

$$\begin{align*}A\subseteq_S B :\equiv \forall x^\Omega \left(x \in A \rightarrow x \in B\right) \end{align*}$$

for $A,B$ of type S and it is straightforward to show that $\subseteq _S$ forms a partial order with respect to equality defined by $=_S$ .

For axioms, we first specify that $\mathrm {eq}$ represents an equivalence relation:

(eq)₁ $$\begin{align} &\forall x^\Omega,y^\Omega(\mathrm{eq} xy\le_0 1), \end{align}$$

(eq)₂ $$\begin{align} &\forall x^\Omega,y^\Omega,z^\Omega \left(x=_\Omega x\land \left( x=_\Omega y \to y=_\Omega x\right)\land \left(x=_\Omega y\land y=_\Omega z\to x=_\Omega z\right)\right). \end{align}$$

Further, we axiomatize that $\in $ , as a relation, is bounded by $1$ on all inputs and behaves as an element relation regarding the operations of union and complement as well as with respect to the empty set:

(∈)₁ $$\begin{align} &\forall x^\Omega \forall A^S (\in x A \le_0 1), \end{align}$$

(∈)₂ $$\begin{align} &\forall x^\Omega (x \not\in \emptyset), \end{align}$$

(∈)₃ $$\begin{align} &\forall x^\Omega \forall A^S, B^S(x \in A \cup B \leftrightarrow x \in A \lor x \in B),\end{align}$$

(∈)₄ $$\begin{align} &\forall x^\Omega \forall A^S (x \in A^c \leftrightarrow x \not\in A). \end{align}$$

Based on the fact that inclusions of elements $x^\Omega $ in elements $A^S$ as facilitated by $\in $ are quantifier-free assertions, the above axioms are (generalized) $\Pi _1$ -sentences and so they are in particular immediately admissible in the context of bound extraction theorems based on the Dialectica interpretation (as will be discussed later in more detail).

We now begin by showing some basic properties of the above operations on algebras provable in this system $\mathcal {F}^\omega $ which, for one, amount to deriving the essential algebraic properties of S as a subalgebra of the full Boolean algebra of the power set of $\Omega $ . Further, for another, all algebraic operations on S behave in a provably extensional way.

Using the recursor constants of the underlying language of $\mathcal {F}^\omega $ in combination with the union operation $\cup $ immediately allows one to also talk about arbitrary finite unions. Concretely, given a sequence of events $A^{S(0)}$ and two natural numbers $n^0\leq _0 m^0$ , we use the abbreviation

$$\begin{align*}\bigcup_{i=n}^{m}A(i):=R_S(m-n,A(n),\lambda B,x.(B\cup A(n+x+1))) \end{align*}$$

where $R_S$ is a (single) type S recursor constant. For $m<_0n$ , we simply set $\bigcup _{i=n}^{m}A(i):=\emptyset $ . We then dually write

$$\begin{align*}\bigcap_{i=n}^mA(i):=\left(\bigcup_{i=n}^m(A(i))^c\right)^c. \end{align*}$$

It is easy to show by induction that the previous extensionality result for $\cup $ extends to these finite unions.

4 Systems for contents on algebras of sets

We now augment the previous system $\mathcal {F}^\omega $ for the treatment of algebras so that we arrive at a system suitable for treating proofs from the theory of probability contents in the sense of the following definition:Footnote ¹⁰

We mainly denote probability contents by the symbol $\mathbb {P}$ . Again, we mainly refer to [Reference Bhaskara Rao and Bhaskara Rao6] as a standard reference for the theory of contents.

The concrete approach that we now take for a formal system for probability contents on algebras is to introduce an additional constant

• • $\mathbb {P}$ of type $1(S)$

to the language of the system $\mathcal {F}^\omega $ . The first defining properties of $\mathbb {P}$ as a probability content are then easily formalized in the underlying language as

(ℙ)₁ $$\begin{align} & \forall A^S (0 \le_{\mathbb{R}} \mathbb{P}(A) \le_{\mathbb{R}} 1), \end{align}$$

(ℙ)₂ $$\begin{align} & \mathbb{P}(\emptyset) =_{\mathbb{R}} 0. \qquad\quad\qquad\end{align}$$

These statements $(\mathbb {P})_1$ and $(\mathbb {P})_2$ are again purely universal statements and therefore immediately admissible in the context of metatheorems based on the (monotone) functional interpretation.

The last property of $\mathbb {P}$ , that is, additivity, if formalized naively via

$$\begin{align*}\forall A^S, B^S(A\cap B=_S\emptyset\to\mathbb{P}(A \cup B) =_{\mathbb{R}} \mathbb{P}(A) + \mathbb{P}(B)), \end{align*}$$

is not purely universal based on the internal definition of $=_S$ and is instead equivalent over $\mathcal {F}^\omega $ (extended with the constant $\mathbb {P}$ ) to the following generalized $\Pi _3$ -sentence:

$$\begin{align*}\forall A^S, B^S\exists x^\Omega(x\not\in A\cap B\to\mathbb{P}(A \cup B) =_{\mathbb{R}} \mathbb{P}(A) + \mathbb{P}(B)). \end{align*}$$

Similar to how the class of so-called type $\Delta $ sentences is treated in, for example, [Reference Günzel and Kohlenbach16], it is clear that this statement would be admissible in the context of bound extraction theorems based on the monotone Dialectica interpretation if the x could be conceived of as being bounded in a suitable sense relative to A and B. Now, as a matter of fact, a crucial perspective for our formal approach to deriving bound extraction theorems for these systems for algebras and probability contents will be that the whole space $\Omega $ can be naturally regarded as uniformly bounded. In the context of a corresponding suitable extension of the notion of majorizability to $\Omega $ which reflects this perspective via assuming that there is a uniform majorant for all $x^\Omega $ , the above axiom actually has a trivial monotone functional interpretation and is thus admissible in the context of the approach to proof mining metatheorems via such a variant of the Dialectica interpretation. This will be discussed in full formal detail later on so that here, for now, we are content with just considering quantification over $\Omega $ as “bounded” and so as “proof-theoretically harmless.”

However, if we were to admit the above sentence as the sole axiom, we would be tasked with deriving all the other properties of $\mathbb {P}$ from this axiom, including monotonicity and thus extensionality of the content as a function, which would require many subtle manipulations of various equalities using the quantifier-free extensionality rule. We therefore instead opt for the following axiomatization which eases the formal development of these properties in the resulting system: For one, instead of additivity, we include the following generalized additivity law which holds for probability contents on algebras:

(ℙ)₃ $$\begin{align} \forall A^S, B^S(\mathbb{P}(A \cup B) =_{\mathbb{R}} \mathbb{P}(A) + \mathbb{P}(B)-\mathbb{P}(A\cap B)). \end{align}$$

This statement is purely universal and thus immediately admissible in the context of our approach to bound extraction theorems as before. The other property of $\mathbb {P}$ that we then axiomatically add is that of the monotonicity of $\mathbb {P}$ , that is,

$$\begin{align*}\forall A^S,B^S\left( A\subseteq_S B\rightarrow \mathbb{P}(A)\leq_{\mathbb{R}}\mathbb{P}(B)\right). \end{align*}$$

Similar to the above, this statement is equivalent to the following (generalized) $\Pi _3$ -statement

(ℙ)₄ $$\begin{align} \forall A^S,B^S\exists x^\Omega\left(\mathbb{P}(A)>_{\mathbb{R}}\mathbb{P}(B)\to x\in A\land x\not\in B\right) \end{align}$$

which in the context of the previously sketched extended notion of majorizability will later have a trivial monotone functional interpretation and thus be admissible in the context of our approach to proof mining metatheorems.

Adding these statements as axioms to the underlying system for algebras of sets, we derive the following system for probability contents on such algebras:

We now begin with some immediate properties of $\mathbb {P}$ provable in the above system.

Contents on algebras enjoy certain continuity properties similar to continuity from above and below for measures but without the existence of limiting sets, that is, infinite unions, etc. (see, e.g., [Reference Bhaskara Rao and Bhaskara Rao6]) and we now discuss how the system $\mathcal {F}^\omega [\mathbb {P}]$ recognizes Cauchy-variants of these properties.

For that, we introduce the following operation on terms of type $S(0)$ that allows for the implicit quantification over a disjoint countable family of sets: given $A^{S(0)}$ , we set $(A\!\uparrow )(0)=A(0)$ and

$$\begin{align*}(A\!\uparrow)(n+1):=A(n+1)\cap \left(\bigcup^n_{i=0} A(i)\right)^c. \end{align*}$$

This operation thus turns A into a sequence of disjoint sets $A\!\uparrow $ with the same (partial) union(s) and if A was already a disjoint family, then it is left unchanged by the operation.

We now begin with a Cauchy-type form of $\sigma $ -additivity of $\mathbb {P}$ as a content. For this, note that for a given $A^{S(0)}$ , the sequence of partial sums

$$\begin{align*}\sum^n_{i=0}\mathbb{P}((A\!\uparrow)(i))=\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^n(A\!\uparrow)(i)\right)=\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^nA(i)\right) \end{align*}$$

is a monotone and bounded sequence of real numbers and thus is Cauchy. The following result that (already a weak fragment of) $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ suffices to prove the Cauchy formulation of the convergence of monotone and bounded sequences is well known:

Lemma 4.4 (folklore, see essentially [Reference Kohlenbach26]).

The system $\textsf {WE}\mbox {-}\textsf {PA}^\omega $ (and actually already a weak fragment thereof) proves that

$$ \begin{align*} &\forall a^{1(0)}\Big(\forall n^0\left(0\le_{\mathbb{R}} a(n)\le_{\mathbb{R}} 1\land a(n) \le_{\mathbb{R}} a(n+1)\right)\\ &\qquad\to \forall k^0\exists N^0\forall n^0,m^0\ge_0 N\left(\vert a(n)-a(m)\vert <_{\mathbb{R}} 2^{-k}\right)\Big). \end{align*} $$

So, instantiating the above result with $a(n)=\sum ^n_{i=0}\mathbb {P}((A\!\uparrow )(i))$ , we can derive that $\mathcal {F}^\omega [\mathbb {P}]$ (and actually already a weak fragment thereof) can prove the Cauchy property of sequences of contents of increasing disjoint unions:

Proposition 4.5. The system $\mathcal {F}^\omega [\mathbb {P}]$ (and actually already a weak fragment thereof) proves

$$\begin{align*}\forall A^{S(0)}\forall k^0\exists N^0\forall n^0,m^0\ge_0 N\left( \left\vert \sum_{i=0}^n\mathbb{P}((A\!\uparrow)(i)) - \sum_{i=0}^m\mathbb{P}((A\!\uparrow)(i))\right\vert<_{\mathbb{R}} 2^{-k}\right). \end{align*}$$

From this Proposition 4.5, we can then immediately derive the following continuity theorems for contents.

5 Systems for $\sigma $ -algebras and probability measures

If we now further require the closure of the underlying algebra of sets under countable unions, we arrive at the notion of a $\sigma $ -algebra which forms the algebraic basis for probability measures.

The requirement that a content on a $\sigma $ -algebra is also well-behaved w.r.t. these countable unions then leads to the notion of a measure on such an algebra.

Definition 5.2 (Measure).

Let $\Omega $ be a set and $S\subseteq 2^\Omega $ be a $\sigma $ -algebra. A measure on S is a content $\mu :S\to [0,\infty ]$ that is also $\sigma $ -additive, that is,

$$\begin{align*}\mu\left(\bigcup_{n=0}^\infty A_n\right)=\sum_{n=0}^\infty\mu(A_n) \end{align*}$$

for any $(A_n)\subseteq S$ with $A_i\cap A_j=\emptyset $ for $i\neq j$ .

The map $\mu $ is called a probability measure if $\mu (\Omega )=1$ . In that case, $(\Omega ,S,\mu )$ is called a probability space.

In this section, we will now discuss how the previous system for algebras $\mathcal {F}^\omega $ and its extension $\mathcal {F}^\omega [\mathbb {P}]$ for treating probability contents can be augmented by a certain intensional treatment of countably infinite unions to provide an apt and tame formal basis for these notions.

5.1 Treating infinite unions tamely

Concretely, to treat countably infinite unions over algebras of sets tamely, we now extend the previous system $\mathcal {F}^\omega $ with the following further constant

• • $\bigcup $ of type $S(S(0))$ ,

providing a term of type S for the resulting union of the sequence of sets coded by the input of type $S(0)$ . So, in the context of suitable axioms specifying that $\bigcup A$ for a given $A^{S(0)}$ represents the union of all $A(n)$ , we can formally induce that the algebra S is closed under these countable unions. The immediate axioms specifying the property that $\bigcup A$ is the corresponding union are

(⋃)₁ $$\begin{align} \forall A^{S(0)}\forall n^0\left( A(n)\subseteq_S \bigcup A\right) \end{align}$$

as well as

$$\begin{align*}\forall A^{S(0)}\forall B^S\left( \forall n^0 \left( A(n)\subseteq_S B\right)\to \bigcup A\subseteq_S B\right), \end{align*}$$

specifying that $\bigcup $ is the join of the elements $A(n)$ in S (seen as a lattice). The first statement ${({\bigcup })}_1$ is immediately admissible in the context of the Dialectica interpretation as it is purely universal. The latter statement is naturally not admissible as is in the context of the usual approach to proof mining metatheorems as it contains a negative universal quantifier of type $0$ that cannot be majorized (which, after all, is also why a uniform variant of arithmetical comprehension is necessary to fully define countable unions of sets, see, e.g., [Reference Kreuzer35] for further discussions).

Since we want to avoid this strong form of comprehension as to not in general distort the strength of the extracted bounds to be able to a priori guarantee the extractability of proof-theoretically tame bounds from proofs, we opt for the next best thing we can do and instead specify the union only intensionally by adding the following rule-variant of the above converse implication

(⋃)₂ $$\begin{align} \frac{F_{qf}\to \forall n^0\left( A(n)\subseteq_SB\right)}{F_{qf}\to \bigcup A\subseteq_S B}\end{align}$$

where A is a term of type $S(0)$ , B is a term of type S and $F_{qf}$ is a quantifier-free formula. So: If $A(n)$ is provably bounded above by B w.r.t. $\subseteq _S$ under some quantifier-free assumptions, then also $\bigcup A$ is provably bounded above by B w.r.t. $\subseteq _S$ under the same assumptions.

Definition 5.3. We write $\mathcal {F}^\omega [\bigcup ]$ for the system resulting from $\mathcal {F}^\omega $ by extending it with the constant $\bigcup $ together with the axiom ${({\bigcup })}_1$ and the rule ${({\bigcup })}_2$ . Similarly, we write $\mathcal {F}^\omega [\bigcup ,\mathbb {P}]$ for the system that arises from $\mathcal {F}^\omega [\mathbb {P}]$ by adding the same constants, axioms, and rules.

Using this intensional approach to countable unions, we can also immediately provide an intensional treatment of countable intersections. For this, we first define $A^c$ for an $A^{S(0)}$ by setting $A^c(n):= A(n)^c$ for any $n^0$ . Using this notation, we then define the countable intersection of a collection of sets represented by an $A^{S(0)}$ via

$$\begin{align*}\bigcap A := \left(\bigcup A^c\right)^c. \end{align*}$$

We then get that analogs of the axiom ${({\bigcup })}_1$ and the rule ${({\bigcup })}_2$ , formulated appropriately for the intersection, are provable in our system $\mathcal {F}^\omega [\bigcup ]$ :

Lemma 5.4. The following statement is provable in $\mathcal {F}^\omega [\bigcup ]$ :

$$\begin{align*}\forall A^{S(0)}\forall n^0\left( \bigcap A\subseteq_S A(n)\right). \end{align*}$$

Further, in $\mathcal {F}^\omega [\bigcup ]$ the following rule is derivable:

$$\begin{align*}\frac{F_{qf}\to \forall n^0\left( B\subseteq_S A(n)\right)}{F_{qf}\to B\subseteq_S \bigcap A}. \end{align*}$$

Proof. For the provability of the first statement, let $x\in \bigcap A$ . By definition we have $x\not \in \bigcup A^c$ . Then by axiom ${({\bigcup })}_1$ , we get $x\not \in A(n)^c$ , that is, $x\in A(n)$ for any n.

Now, for the rule, suppose that we provably have $F_{qf}\to \forall n\left ( B\subseteq A(n)\right )$ . Then we also provably have $F_{qf}\to \forall n(A(n)^c\subseteq B^c)$ and using the rule ${({\bigcup })}_2$ , we get $F_{qf}\to \bigcup A^c\subseteq B^c$ . Thus also $F_{qf}\to B\subseteq \left (\bigcup A^c\right )^c=\bigcap A$ as desired.

5.2 Handling probability measures

As we have seen in Propositions 4.5 and 4.6, the system $\mathcal {F}^\omega [\mathbb {P}]$ already provides Cauchy-variants of the convergence of monotone sequences of events as well as of sums of disjoint events. In the theory of measures on $\sigma $ -algebras, the resulting limits of course correspond to the measure of respective infinite unions or intersections. Thus, the natural question of whether and how this can be formally represented in the system $\mathcal {F}^\omega [\bigcup ,\mathbb {P}]$ immediately arises. And while for a disjoint family represented by $A^{S(0)}$ , the limit

$$\begin{align*}0\leq \mathbb{P}\kern1.5pt\left(\bigcup A\right)-\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^nA(i)\right)\to 0\text{ for }n\to\infty \end{align*}$$

holds true, there is in general no computable rate of convergence for this expression in the sense that even a function $\varphi $ of type $1$ such that

(∘) $$ \begin{align} \forall k^0\exists n\leq_0 \varphi(k)\left(\mathbb{P}\kern1.5pt\left(\bigcup A\right)-\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^nA(i)\right)\leq_{\mathbb{R}} 2^{-k}\right) \end{align} $$

is in general not computable (see Remark 5.5 for an example). Nevertheless, we want to point out that while therefore the convergence of the sequence $\sum _{i=0}^n\mathbb {P}(A(i))$ towards $\mathbb {P}(\bigcup A)$ cannot be provable in any system that allows for the extraction of computable and uniform bounds, the system $\mathcal {F}^\omega [\bigcup ,\mathbb {P}]$ still provides an intensional version of that convergence in the following sense:

1. 1. By Proposition 4.5, the sequence of partial sums $\sum _{i=0}^n\mathbb {P}(A(i))$ is provably Cauchy.

2. 2. Using the additivity and monotonicity of $\mathbb {P}$ , that is, axioms $(\mathbb {P})_3$ and $(\mathbb {P})_4$ , we get by $\bigcup _{i=0}^nA(i)\subseteq _S \bigcup A$ that

   $$\begin{align*}\sum_{i=0}^n\mathbb{P}(A(i))=_{\mathbb{R}}\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^n A(i)\right)\leq_{\mathbb{R}} \mathbb{P}\kern1.5pt\left(\bigcup A\right) \end{align*}$$
   holds provably.

3. 3. For any object $B^S$ such that we provably have $\forall n^0\left (A(n)\subseteq _S B\right )$ , we get $\bigcup A\subseteq _S B$ using the rule ${({\bigcup })}_2$ and so we get provably

   $$\begin{align*}\mathbb{P}\kern1.5pt\left(\bigcup A\right)\leq_{\mathbb{R}} \mathbb{P}(B) \end{align*}$$
   in that case by monotonicity of $\mathbb {P}$ .

So the value $\mathbb {P}(\bigcup A)$ is at least intensionally specified to be the limit of the partial sums $\sum _{i=0}^n\mathbb {P}(A(i))$ as $\mathbb {P}(\bigcup A)$ is bounded below by this nondecreasing sequence of partial sums and intensionally bounded above by the probability of any set which provably sits above the given partial unions $\bigcup _{i=0}^nA(i)$ .

The case that we want to make is now twofold: For one, as mentioned in the introduction, the theory of contents already exhausts large parts of the theory of measures in the sense that often already the properties of contents on algebras suffice to carry out proofs for properties of measures on $\sigma $ -algebras (as will also be the case in the applications discussed later). For another, we want to argue that even in situations where one cannot do just with finite unions and content, such an intensional specification of countable unions and their measures might suffice for formalizing a given proof and all the while guaranteeing the extractability of tame bounds a priori. If that is not the case, then the result under consideration might be considered to be inherently “untame” and a full treatment of the comprehension principle needed to define the respective unions could be necessary. We therefore regard $\mathcal {F}^\omega [\bigcup ,\mathbb {P}]$ as a suitable tame base system for treating probability measures on $\sigma $ -algebras.

Remark 5.5. For an example where $\varphi $ in ( $\circ $ ) is not computable, we take $(r_i)\subseteq (0,1)$ to be a sequence of computable real numbers such that

$$\begin{align*}a_n=\sum_{i=0}^nr_i\to a \le 1 \end{align*}$$

without a computable rate of convergence.Footnote ¹¹ We now define $\Omega =\mathbb {N}\cup \{\infty \}$ as well as $S=2^\Omega $ . On the discrete sample space $\Omega $ , we then define a probability mass function p via $p(i)=r_{i}$ for $i\in \mathbb {N}$ as well as $p(\infty )=1-a$ . This p then as usual induces a probability measure $\mathbb {P}$ on the $\sigma $ -algebra S defined for $A\subseteq \Omega $ via

$$\begin{align*}\mathbb{P}(A)=\sum_{a\in A}p(a). \end{align*}$$

Clearly $(\Omega ,S,\mathbb {P})$ is a probability space where

$$\begin{align*}\mathbb{P}(\mathbb{N})-\sum_{i=1}^{n+1}\mathbb{P}(\{i\})=a-\sum_{i=0}^nr_i=a-a_n \end{align*}$$

cannot have a computable rate of convergence to $0$ .

6 Intensional intervals, inverse mappings and measurable functions

In this section, we now extend the machinery of the previous logical systems so that we are able to deal with functions $f:\Omega \to \mathbb {R}$ that are measurable in a certain suitable sense relative to algebras. As such, the treatment given here will be instrumental for our approach to integrable functions in Section 7, for the applications discussed in Section 9, and for the proof-theoretic transfer principles for implications between modes of convergence in Section 10. For this, we now first recall the essential definitions and basic results.

We refer to [Reference Halmos17] as a standard reference for Borel $\sigma $ -algebras in particular and measure theory in general (in particular regarding the well-definedness of the above definition for which one needs to see that the intersection of any family of ( $\sigma $ -)algebras is again a ( $\sigma $ -)algebra).

Crucial for us will be the notion of a generating set of a ( $\sigma $ -)algebra.

In that terminology, the Borel $\sigma $ -algebra $\mathcal {B}(X)$ is the $\sigma $ -algebra generated by the open subsets of the underlying topological space.

For the special case of the real numbers as a topological space with the usual topology induced by the metric distance, we in particular get the following canonical generators besides the open subsets of $\mathbb {R}$ .

One then arrives at the notion of a measurable function which we for simplicity only formulate for real-valued functions here.

Definition 6.4 (Borel-measurable function).

Let $(\Omega ,S,\mu )$ be a content space. A function $f:\Omega \to \mathbb {R}$ is called Borel-measurable if

$$\begin{align*}f^{-1}(B):=\{x\in \Omega\mid f(x)\in B\}\in S \end{align*}$$

for all $B\in \mathcal {B}(\mathbb {R})$ .

As we will crucially use later on, Borel-measurability is simply characterized by a similar condition on a generating set.

If the underlying algebra S is not a $\sigma $ -algebra and if $\mu $ is only a content, then the requirement that the preimages of the above collection of intervals are included is still statable but might result in something weaker than full Borel-measurability. We call a function $f:\Omega \to \mathbb {R}$ where the preimages of all intervals $[a,b]$ for $a,b\in \mathbb {R}$ are included in a corresponding algebra $S\subseteq 2^\Omega $ to be weakly Borel-measurable. It will in particular be this notion of weakly Borel-measurable functions that we will rely on later in the context of our approach towards Lebesgue integrals for probability contents. It should be noted that by requiring the inclusion $f^{-1}([a,b])\in S$ for two real numbers $a,b\in \mathbb {R}$ and an algebra S, we in particular also obtain that $f^{-1}([a,b))=f^{-1}([a,b])\cap \left (f^{-1}([b,b])\right )^c\in S$ as S is an algebra.Footnote ¹²

To formally deal with the notion of (weak) Borel-measurability, we thus need access to the collection of the closed intervals $[a,b]$ for $a,b\in \mathbb {R}$ generating the Borel-algebra. For this, we will introduce an intensional approach to real intervals in the next subsection to provide formal means of operating on these generators. These intensional variants of real intervals can then be processed by a general type of inverse map using which we will be able to state the measurability of a function formally.

6.1 Intensional Intervals

Concretely, we now provide a quantifier-free (and thus in a way intensional) account of the closed intervals $[a,b]$ (and thus also of the half-open intervals $[a,b)$ as discussed above) by introducing a further constant to the language:

• • $[\cdot ,\cdot ]$ of type $0(1)(1)(1)$ .

Given two inputs $a^1, b^1$ , this function shall return a characteristic function for an intensional representation of the corresponding interval. For this, we add the following axioms:

([⋅,⋅])₁ $$\begin{align} &\forall a^1,b^1,r^1\left([a,b](r)\leq_0 1\right), \end{align}$$

([⋅,⋅])₂ $$\begin{align} &\forall a^1,b^1,r^1\left(r\in [a,b]\to a\leq_{\mathbb{R}} r\leq_{\mathbb{R}}b\right), \end{align}$$

([⋅,⋅])₃ $$\begin{align} &\forall a^1,b^1,r^1\left(a<_{\mathbb{R}}r<_{\mathbb{R}} b\to r\in [a,b]\right), \end{align}$$

([⋅,⋅])₄ $$\begin{align} &\forall a^1,b^1\left(a,b\in [a,b]\right). \end{align}$$

Here, we wrote $[a,b]$ for $[\cdot ,\cdot ]ab$ as well as $r\in [a,b]$ for $[a,b](r)=_00$ . Note that this is an intensional representation of the set as we have $a,b\in [a,b]$ but we cannot conclude from $r=a$ or $r=b$ that $r\in [a,b]$ .

We also introduce the following notation used later: if we are given a system $\mathcal {C}^\omega $ , we write $\mathcal {C}^\omega [\mathrm {Int}]$ to denote the extension of $\mathcal {C}^\omega $ by the above constant $[\cdot ,\cdot ]$ and the axioms $([\cdot ,\cdot ])_1$ – $([\cdot ,\cdot ])_4$ for treating the closed intervals.

In similarity to the discussion above, these closed intervals then also provide a quantifier-free access to the half-open intervals in the following way: We define $[\cdot ,\cdot )$ of type $0(1)(1)(1)$ via

$$\begin{align*}[\cdot,\cdot):=\lambda a^1,b^1,r^1.\max\left\{[a,b](r),1-[b,b](r)\right\}. \end{align*}$$

Also for $[\cdot ,\cdot )$ , we write $[a,b)$ for $[\cdot ,\cdot )ab$ as well as $r\in [a,b)$ for $[a,b)(r)=_00$ .

In the context of that definition, we in particular obtain relatively immediately that $[\cdot ,\cdot )$ defined as such satisfies properties that intensionally specify the half-open intervals similar to how we have specified closed intervals above with the axioms $([\cdot ,\cdot ])_1$ – $([\cdot ,\cdot ])_4$ , that is, we for one have $a\in [a,b)$ but from $r=a$ , we cannot infer $r\in [a,b)$ and we have $b\not \in [a,b)$ but from $r=b$ , we cannot infer $r\not \in [a,b)$ . This is collected in the following lemma:

Lemma 6.6. The system $\mathcal {A}^\omega [\mathrm {Int}]$ proves the following properties of $[\cdot ,\cdot )$ :

1. 1. $\forall a^1,b^1,r^1\left ([a,b)(r)\leq _0 1\right )$ ,

2. 2. $\forall a^1,b^1,r^1\left (r\in [a,b)\to a\leq _{\mathbb {R}} r<_{\mathbb {R}}b\right )$ ,

3. 3. $\forall a^1,b^1,r^1\left (a<_{\mathbb {R}}r<_{\mathbb {R}} b\to r\in [a,b)\right )$ ,

4. 4. $\forall a^1,b^1\left (a<_{\mathbb {R}}b\to a\in [a,b)\land b\not \in [a,b)\right )$ .

We omit the proof as it is rather immediate.

6.2 The inverse map

We now provide a treatment of the inverse map for a given function f of type $1(\Omega )$ . For this, we actually introduce a uniform operator into the language via a constant

• • $(\cdot )^{-1}$ of type $0(\Omega )(0(1))(1(\Omega ))$

that provides an inverse map for any given function $f^{1(\Omega )}$ in the sense that, writing $f^{-1}$ for $(\cdot )^{-1}f$ , the functional $f^{-1}$ receives a subset of the reals coded via a characteristic function $A^{0(1)}$ and maps this into a characteristic function $f^{-1}A$ of type $0(\Omega )$ coding a subset of the underlying space $\Omega $ .

This type of map is then governed by the following two axioms:

(Inv)₁ $$\begin{align} &\forall f^{1(\Omega)}\forall A^{0(1)}\forall x^\Omega\left(f^{-1}Ax\leq_0 1\right), \end{align}$$

(Inv)₂ $$\begin{align}&\forall f^{1(\Omega)}\forall A^{0(1)}\forall x^\Omega\left( x\in f^{-1}A\leftrightarrow f(x)\in A\right). \end{align}$$

Here, we wrote $f(x)\in A$ for $Af(x)=_00$ and $x\in f^{-1}A$ for $f^{-1}Ax=_00$ .

Also for this type of extension, we introduce the following generic notation: given a system $\mathcal {C}^\omega $ , we write $\mathcal {C}^\omega [\mathrm {Inv}]$ to denote the extensions of $\mathcal {C}^\omega $ by the constant $(\cdot )^{-1}$ and the above axioms for treating the inverse map.

6.3 Measurability of functions

In the context of the intensional representations for closed intervals, generating the Borel $\sigma $ -algebra on the reals, as well as using the general inverse map introduced before, we are now in the position of formulating the (weak) Borel-measurability of a function $f^{1(\Omega )}$ formally in the underlying language by

$$\begin{align*}\forall a^1,b^1\exists A^S\forall x^\Omega\left( x\in f^{-1}([a,b])\leftrightarrow x\in A\right). \end{align*}$$

The inner matrix (based on the fact that stating element relations via $\in $ is quantifier-free and as we use the abbreviation $x\in f^{-1}([a,b])$ for $f^{-1}[a,b]x=_00$ as introduced in Section 6.2) is quantifier-free and thus the above sentence is a generalized $\Pi _3$ -statement. Similar to the monotonicity statement of the content $\mathbb {P}$ , this statement would be admissible in the context of the monotone Dialectica interpretation if the quantification over elements of type S could be conceived of as being bounded in a suitable sense. Similar to how we argued with the monotonicity of $\mathbb {P}$ , a crucial perspective for our formal approach to bound extraction theorems later on will be that, besides the whole space $\Omega $ , we will also be able to regard the space S as uniformly bounded, formally encapsulated via a corresponding suitable extension of the notion of majorizability to S used later. In that context, such a sentence has a trivial monotone functional interpretation and we will use this later to formulate admissible axioms stating that certain classes of functions are indeed measurable.

7 Treating integration over probability contents

We now want to extend the previous system $\mathcal {F}^\omega [\mathbb {P}]$ for probability contents on algebras so that we can treat a certain class of integrable functions $f:\Omega \to \mathbb {R}$ , in particular so that we obtain a suitable base for random variables and their moments as used in various applications, especially those illustrated in Section 9.

For the usual approach to the integral over contents, which mimics that of the Lebesgue integral, we mainly follow the exposition given in [Reference Bhaskara Rao and Bhaskara Rao6] (where the corresponding notion is introduced under the name of the “D-integral”) which we detail here to some degree to provide the necessary mathematical basics for the axiomatizations chosen later. Concretely, let $\Omega $ be a set and S an algebra on it and let $\mu $ be a finite content on S (i.e., $\mu (\Omega )<+\infty $ ). One then first arrives at a notion of simple function that is completely analogous to how it is usually defined in the context of measure theory, that is, a simple function is a function $f:\Omega \to \mathbb {R}$ of the form

$$\begin{align*}f(x)=\sum_{i=0}^n b_i\chi_{A_i} \end{align*}$$

for given sets $A_i\in S$ and values $b_i\in \mathbb {R}$ .Footnote ¹³ For such a function f, the integral over $\mu $ is simply defined as

$$\begin{align*}\int{f}\,\mathrm{d}\mu=\sum_{i=0}^n b_i\mu(A_i), \end{align*}$$

also in similarity to usual Lebesgue integrals over measures. A general function $f:\Omega \to \mathbb {R}$ is now declared integrable if there is a sequence of simple functions $f_n$ such that (1) the $f_n$ converge to f in a suitable senseFootnote ¹⁴ and (2) the sequence satisfies

$$\begin{align*}\lim_{n,m\to\infty}\int{\vert f_n-f_m\vert}\,\mathrm{d}\mu=0. \end{align*}$$

Such a sequence is called a determining sequence for f and using such a determining sequence, one then defines the integral of f via

$$\begin{align*}\int{f}\,\mathrm{d}\mu=\lim_{n\to\infty}\int{f_n}\,\mathrm{d}\mu. \end{align*}$$

Crucially, this limit is well-defined as the following general result shows:

Lemma 7.1 (Lemma 4.4.12 in [Reference Bhaskara Rao and Bhaskara Rao6]).

Let f be an integrable function and let $(f_n)$ be a determining sequence for f. Then $f_n-f$ is integrable and

$$\begin{align*}\lim_{n\to\infty}\int{\vert f_n-f\vert}\,\mathrm{d}\mu=0. \end{align*}$$

This notion of a (D-)integral defined for contents then shares many of the familiar properties of Lebesgue integrals defined for measures. One particularly useful property is that every integrable function f is measurable in the following extended sense, called $T_2$ -measurable in [Reference Bhaskara Rao and Bhaskara Rao6]: for any $\varepsilon>0$ , there is a partition $A_0,\dots , A_n\in S$ of $\Omega $ such that $\mu (A_0)<\varepsilon $ and $\vert f(x)-f(y)\vert <\varepsilon $ for any $x,y\in A_i$ and any i. In particular, for any function f that is measurable in this sense, one gets that f is integrable if, and only if, $\vert f\vert $ is integrable (see, e.g., Corollary 4.4.19 in [Reference Bhaskara Rao and Bhaskara Rao6]).

Now, a major part of the theory of integrals over contents (like, e.g., a nice correspondence between the so-called D- and S-integrals, the latter being similar in spirit to a Riemann-Stieltjes integral, and the fact that the notions of $T_2$ -measurability and integrability coincide as shown by Theorem 4.5.7 in [Reference Bhaskara Rao and Bhaskara Rao6], among many others) depends on the assumption that the integrated functions are bounded. In that way, we will similarly require that all integrated functions are bounded.

In fact, using the proof-theoretic perspective of the approach taken here, we find that this assumption of the boundedness of functions is also suggested as a necessity in our formal approach by the notion of majorizability employed later. Concretely, as discussed before, to develop a proof-theoretically tame theory of algebras and contents, we have regarded $\Omega $ and S as uniformly bounded in the sense that we later regard all elements of these spaces as uniformly majorized by the content of the full space $\mu (\Omega )$ , that is, by $1$ in the context of a probability content, which we denote in writing by $1\gtrsim _\Omega x$ and $1\gtrsim _S A$ for $x\in \Omega $ and $A\in S$ . While this will be discussed comprehensively and in full formal detail later, we here already look at what this definition entails for majorizable functions f of type $1(\Omega )$ : a function $f^*$ of type $0(0)(0)$ is a majorant for f, written $f^*\gtrsim _{1(\Omega )} f$ , if

$$\begin{align*}f^*(m)(k)\geq f^*(n)(j)\geq f(x)(i)\text{ whenever }m\geq n\gtrsim_\Omega x\text{ and }k\geq j\geq i. \end{align*}$$

Therefore, as $1\gtrsim _\Omega x$ for any x, we in particular derive that

$$\begin{align*}\vert f(x)\vert\leq [\vert f(x)\vert](0)+1\leq f(x)(0)+1\leq f^*(1)(0)+1 \end{align*}$$

for any x, using the monotonicity of the coding of rational numbers (see, e.g., the discussion on p.430 in [Reference Kohlenbach26]). Thus, the real number represented by $f(x)$ is uniformly bounded by $f^*(1)(0)+1$ for any x and so any majorizable function of type $1(\Omega )$ represents a bounded function $\Omega \to \mathbb {R}$ . In other words, boundedness of integrable functions is suggested as a necessary assumption by the chosen proof-theoretic methodology.

Lastly, we will actually require that the integrated functions are not only $T_2$ -measurable in the sense discussed above but that they even are weakly Borel-measurable in the sense discussed before (i.e., that the preimages of closed intervals are included in the underlying algebra). Clearly, any bounded and weakly Borel-measurable function is also $T_2$ -measurable in the previous sense and thus integrable as discussed before. While this class is slightly restricted compared to that of all integrable functions, we find that it has two central advantages: For one, it allows for a particularly smooth and proof-theoretically tame approach to the integral that is amenable to bound extraction theorems. For another, in virtually all previous ad hoc proof mining applications to measure and probability theory involving integrals, already the stronger (or, in the context of $\sigma $ -algebras, equivalent) property of being Borel-measurable is assumed so that a treatment of this class still allows for our metatheorems established later to provide a proof-theoretic explanation of the respective extractions. Moreover, this is in particular true for the applications discussed later in Section 9. In that way, the approach to the integral via this assumption should be understood to be merely indicative regarding the possibilities of the present formal approach, where various other measurability and integrability notions could similarly be accommodated.

Now, the formal approach to the integral over a probability content is to abstractly encode a suitable subspace of bounded and weakly Borel-measurable functions closed under linear combinations, multiplications with characteristic functions, and absolute valuesFootnote ¹⁵ intensionally via the use of a characteristic function and then to introduce the integral for all functions from this space as well as the relevant closure properties using further constants and axioms. For this, we now initially introduce two further constants

• • I of type $0(1(\Omega ))$ ,

• • $\left \lVert \cdot \right \rVert _\infty $ of type $1(1(\Omega ))$ ,

into the language of $\mathcal {F}^\omega [\mathbb {P},\mathrm {Int},\mathrm {Inv}]$ . The first of these is the previously mentioned characteristic function providing an intensional account of a space closed under linear combinations, multiplications with characteristic functions and absolute values as well as containing only bounded and weakly Borel-measurable functions and the latter is used to formally introduce the supremum norm on these functionals. As initial axioms, we now therefore stipulate the following:

(I)₁ $$\begin{align} &\forall f^{1(\Omega)}\left( If\leq_01\right), \end{align}$$

(I)₂ $$\begin{align} &\forall A^S\left( (\lambda x.(x\in A))\in I\right), \end{align}$$

(I)₃ $$\begin{align} &\forall f^{1(\Omega)},a^1,b^1\exists A^S\forall x^\Omega\left(f\in I\to \left(x\in f^{-1}([a,b])\leftrightarrow x\in A\right)\right), \end{align}$$

(I)₄ $$\begin{align} &\forall f^{1(\Omega)}, x^\Omega\left( f\in I\to \left( \vert f(x)\vert\leq_{\mathbb{R}} \left\lVert f\right\rVert_\infty\right)\right), \end{align}$$

(I)₅ $$\begin{align} &\forall f^{1(\Omega)},k^0\exists x^\Omega\left( f\in I\to \left( \left\lVert f\right\rVert_\infty-2^{-k}\leq_{\mathbb{R}} \vert f(x)\vert\right)\right), \end{align}$$

(I)₆ $$\begin{align} &\forall f^{1(\Omega)},g^{1(\Omega)},\alpha^1,\beta^1\left(f,g\in I\to (\lambda x.(\alpha f(x)+\beta g(x)))\in I\right), \end{align}$$

(I)₇ $$\begin{align} &\forall f^{1(\Omega)}\left(f\in I\to \lambda x.\vert f(x)\vert\in I\right), \end{align}$$

(I)₈ $$\begin{align} &\forall f^{1(\Omega)},A^S\left(f\in I\to \lambda x.\left(f(x)(x\in A)\right)\in I\right). \end{align}$$

The axioms $(I)_3$ and $(I)_5$ will again be admissible later because of the extended notion of majorizability on $\Omega $ and S. Note also that axioms $(I)_4$ and $(I)_5$ together specify $\left \lVert f\right \rVert _\infty $ as the least upper bound on $\vert f(x)\vert $ .Footnote ¹⁶

In the following, we will write $\chi _A$ for the function $\lambda x.(x\in A^c)$ .Footnote ¹⁷ Also, in the following we just briefly write $\alpha f+ \beta g$ for $\lambda x.(\alpha f(x)+\beta g(x))$ as well as $\vert f\vert $ for $\lambda x.(\vert f(x)\vert )$ and $f\chi _A$ for $\lambda x.\left (f(x)\chi _A(x)\right )$ .

As the operations $\max $ and $\min $ can be defined on functions using the absolute value via

$$\begin{align*}\max\{f,g\}=(f+g+\vert f-g\vert)/2\text{ and }\min\{f,g\}=-\max\{-f,-g\}, \end{align*}$$

we immediately get that axiom $(I)_7$ implies the closure of I under these operations and thus we have effectively axiomatized that I in particular is a Riesz space of bounded functions with the respective operations and thus our approach is similar to how abstract integration spaces are approached in the context of Daniell integrals [Reference Daniell7] where a similar collection of functions is presumed.

To deal with the integral, we add a further constant

• • $\int {\cdot }\,\mathrm {d}\mathbb {P}$ of type $1(1(\Omega ))$ .

The first two axioms for the integral are now that it behaves as expected on characteristic functions and that the integral is a linear function:

(∫)₁ $$\begin{align} &\forall A^S\left( \int{\chi_A}\,\mathrm{d}\mathbb{P}=_{\mathbb{R}}\mathbb{P}(A)\right), \end{align}$$

(∫)₂ $$\begin{align} &\forall f^{1(\Omega)}, g^{1(\Omega)},\alpha^1,\beta^1\left( f,g\in I\to \int{(\alpha f+\beta g)}\,\mathrm{d}\mathbb{P}=_{\mathbb{R}} \alpha\int{f}\,\mathrm{d}\mathbb{P}+\beta\int{g}\,\mathrm{d}\mathbb{P}\right). \end{align}$$

Using these two axioms, we immediately get that the integral behaves as expected on simple functions.

The major other statement that we need to axiomatize is that any function in I is actually integrable in the sense that its integral is well-defined and arises as the limit of a sequence of integrals of simple functions. In the context of our standing assumption that our functions are bounded and weakly Borel-measurable, we can in particular derive the following general result that guarantees this property, and inspires the subsequent axiom:

Lemma 7.2 (essentially folklore).

Let $\Omega $ be a set, S an algebra on $\Omega $ and $\mu $ a probability content on S. Let f be a weakly Borel-measurable function and assume that $\vert f\vert $ is bounded by $b\in \mathbb {N}^*$ , that is, $\vert f(x)\vert \leq b$ for all $x\in \Omega $ . For a given k, define

$$\begin{align*}I_{k,i}=\left[-b+\frac{i}{2^k},-b+\frac{i+1}{2^k}\right) \text{ for }i=0,1,\dots,b2^{k+1}-2\text{ and }I_{k,b2^{k+1}-1}=\left[\frac{b2^k-1}{2^k},b\right]. \end{align*}$$

Then:

$$\begin{align*}\forall k\in\mathbb{N}\left(\int{\left\vert f-\sum_{i=0}^{b2^{k+1}-1}\left(-b+\frac{i}{2^{k}}\right)\chi_{f^{-1}(I_{k,i})}\right\vert}\,\mathrm{d}\mathbb{P}\leq 2^{-k}\right). \end{align*}$$

Proof. Let $k\in \mathbb {N}$ . As all $I_{k,i}$ are disjoint and cover $[-b,b]$ and since $\vert f\vert $ is bounded by b, their preimages under f are disjoint and cover $\Omega $ . Thus

$$\begin{align*}1=\mathbb{P}(\Omega)=\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^{b2^{k+1}-1}f^{-1}(I_{k,i})\right)=\sum_{i=0}^{b2^{k+1}-1}\mathbb{P}(f^{-1}(I_{k,i})) \end{align*}$$

and for any $k\in \mathbb {N}$ :

$$\begin{align*}f(x)=\sum_{i=0}^{b2^{k+1}-1} f(x)\chi_{f^{-1}(I_{k,i})}(x). \end{align*}$$

Further, for $x\in f^{-1}(I_{k,i})$ , it clearly holds that

$$\begin{align*}\left\vert f(x)-\left(-b+\frac{i}{2^{k}}\right)\right\vert\leq \frac{1}{2^k}. \end{align*}$$

As k was arbitrary, the function f is integrable by Theorem 4.5.7 in [Reference Bhaskara Rao and Bhaskara Rao6] (use, e.g., the equivalence between (viii) and (v)). Thus, using the monotonicity and linearity of the integral on contents (see, e.g., Theorem 4.4.13 in [Reference Bhaskara Rao and Bhaskara Rao6]), we have

$$ \begin{align*} \int{\left\vert f-\sum_{i=0}^{b2^{k+1}-1}\left(-b+\frac{i}{2^{k}}\right)\chi_{f^{-1}(I_{k,i})}\right\vert}\,\mathrm{d}\mathbb{P}&=\int{\left\vert\sum_{i=0}^{b2^{k+1}-1}\left(f+b-\frac{i}{2^{k}}\right)\chi_{f^{-1}(I_{k,i})}\right\vert}\,\mathrm{d}\mathbb{P}\\&\leq \sum_{i=0}^{b2^{k+1}-1}\int{\left\vert f-\left(-b+\frac{i}{2^{k}}\right)\right\vert\chi_{f^{-1}(I_{k,i})}}\,\mathrm{d}\mathbb{P}\\&\leq \sum_{i=0}^{b2^{k+1}-1}\int{\frac{1}{2^k}\chi_{f^{-1}(I_{k,i})}}\,\mathrm{d}\mathbb{P}\\&\leq \sum_{i=0}^{b2^{k+1}-1}\frac{1}{2^k}\mathbb{P}(f^{-1}(I_{k,i}))\\&\leq \frac{1}{2^k}\sum_{i=0}^{b2^{k+1}-1}\mathbb{P}(f^{-1}(I_{k,i}))=\frac{1}{2^k}.\\[-47pt] \end{align*} $$

To axiomatize the integrability of a function $f\in I$ , it thus suffices to state the conclusion of the above lemma and although we could formalize this directly by employing the general inverse mapping and intensional intervals, we can actually avoid this machinery here at the mild expense of quantifying over the sequence of sets used in the simple functions instead of explicitly specifying them. Concretely, we consider the following third axiomFootnote ¹⁸

(∫)₃ $$\begin{align} \forall f^{1(\Omega)}\forall k^0\exists A^{S(0)}\left(f\in I\to \int{\left\vert f-\sum_{i=0}^{2^{k+1}b_f-1}\left(-b_f+\frac{i}{2^{k}}\right)\chi_{A(i)}\right\vert}\,\mathrm{d}\mathbb{P}\leq_{\mathbb{R}} 2^{-k}\right), \end{align}$$

where we wrote $b_f:=\left \lVert f\right \rVert _\infty (0)+1$ and have used that, since $\left \lVert f\right \rVert _\infty \geq _{\mathbb {R}} \vert f(x)\vert $ for all x, it holds that the natural number $b_f$ similarly bounds $\vert f\vert $ . Again, by the later considerations on majorizability whereby also A of type $S(0)$ can be regarded as uniformly bounded, this axiom will later be admissible in the context of our approach to proof mining metatheorems via the monotone functional interpretation.

Lastly, to also devise a practical system, it will be convenient to also axiomatically include (instead of discussing how it might be provable in the system) that the integral of a positive function $f\in I$ is positive. Naively, this statement can be written as

$$\begin{align*}\forall f^{1(\Omega)}\left( f\in I\land \forall x^\Omega\left( f(x)\geq_{\mathbb{R}} 0\right)\to \int{f}\,\mathrm{d}\mathbb{P}\geq_{\mathbb{R}} 0\right) \end{align*}$$

which is not a priori admissible in the context of our approach to bound extraction theorems due to the hidden negative universal type $0$ quantifier in $\geq _{\mathbb {R}}$ . However, if we rewrite the above statement in the prenexed form

$$\begin{align*}\forall f^{1(\Omega)}\forall k^0\exists x^\Omega\exists j^0\left( f\in I\land \int{f}\,\mathrm{d}\mathbb{P}<_{\mathbb{R}} -2^{-k}\to \left( f(x)\leq_{\mathbb{R}} -2^{-j}\right)\right), \end{align*}$$

we can witness the quantifier over j simply by $j=k$ which can be immediately seen using only very basic properties of the integral (see, e.g., Theorem 4.4.13 in [Reference Bhaskara Rao and Bhaskara Rao6]) so that we arrive at the axiom

(∫)₄ $$\begin{align} \forall f^{1(\Omega)}\forall k^0\exists x^\Omega\left( f\in I\land \int{f}\,\mathrm{d}\mathbb{P}<_{\mathbb{R}} -2^{-k}\to\left( f(x)\leq_{\mathbb{R}} -2^{-k}\right)\right) \end{align}$$

which, in the context of our perspective that we regard quantification over $\Omega $ as bounded, will later be admissible in the context of the monotone functional interpretation.

We want to note that the axioms ${({\int })}_2$ – ${({\int })}_4$ roughly correspond to the four central properties of an abstract “I-integral” in the context of Daniell’s approach to integration [Reference Daniell7] and so, in some sense, our approach to the integral here can be regarded as a sort of effectivized implementation of the Daniell integral.

With all of these constants and axioms, we then arrive at the following system for integrals:

We end this section with some immediate properties of the integral over contents that are provable in our system. A more intricate use of the integrals will then be made in Section 9 where they (together with the boundedness and weak Borel-measurability) feature crucially in the formal explanation of a previous proof-mining application (and in particular highlight the usability of the above axiomatic approach to the integral with regard to the proof mining practice). As is common in proof mining, however, this approach to the integral enjoys a large degree of flexibility in the sense that it can, of course, be immediately augmented by further constants and axioms specifying other properties of the integral that might be crucial in a certain application, as also already mentioned before.

Note that by item (2) of the above lemma together with the axioms on the supremum norm $\left \lVert \cdot \right \rVert _\infty $ , we in particular have that $\int {\cdot }\,\mathrm {d}\mathbb {P}$ is extensional w.r.t. $\left \lVert \cdot \right \rVert _\infty $ in the sense that

$$\begin{align*}\forall f^{1(\Omega)},g^{1(\Omega)}\left( \left\lVert f-g\right\rVert_\infty=_{\mathbb{R}}0\to \left\vert\int{f}\,\mathrm{d}\mathbb{P}-\int{g}\,\mathrm{d}\mathbb{P}\right\vert=_{\mathbb{R}}0\right). \end{align*}$$

8 A bound extraction theorem

We now establish our main results, the bound extraction theorems, for the systems introduced previously. For that, as hinted at in the introduction, we follow the approach of the first metatheorems using abstract types presented in [Reference Kohlenbach25] (see also [Reference Gerhardy and Kohlenbach12, Reference Kohlenbach26]).Footnote ¹⁹ As the outline of our approach is rather standard in that way, we will sometimes only sketch the arguments instead of giving full, detailed proofs, only spelling out those parts that are sensitive to the new ideas introduced in this paper. Throughout, to ease notation, we write $\mathcal {C}^\omega $ for the system $\mathcal {F}^\omega $ or one of its extensions as discussed previously.

As in the works mentioned above, the main tool for the metatheorems presented here is Gödel’s Dialectica interpretation [Reference Gödel13] which is combined with a negative translation by Kuroda [Reference Kuroda36]. We recall the definitions of those central proof interpretations here:

For the combination of these two interpretations, the following soundness result is one of the two central technical tools in the context of the proof of the proof mining metatheorems. In that context, we define $\mathcal {C}^{\omega -}$ as the system $\mathcal {C}^\omega $ without the schemas $\textsf {QF}\mbox {-}\textsf {AC}$ and $\textsf {DC}$ .

Lemma 8.3 (essentially [Reference Kohlenbach25]).

Let $\mathcal {P}$ be a set of universal sentences and let $A(\underline {a})$ be an arbitrary formula (with only the variables $\underline {a}$ free) in the language of $\mathcal {F}^\omega $ . Then the rule

$$\begin{align*}\begin{cases}\mathcal{F}^\omega+\mathcal{P}\vdash A(\underline{a})\Rightarrow\\ \mathcal{F}^{\omega-}+(\mathrm{BR})+\mathcal{P}\vdash\forall\underline{a},\underline{y}(A')_D(\underline{t}\underline{a},\underline{y},\underline{a})\end{cases} \end{align*}$$

holds where $\underline {t}$ is a tuple of closed terms of $\mathcal {F}^{\omega -}+(\mathrm {BR})$ which can be extracted from the respective proof and $(\mathrm {BR})$ is the schema of simultaneous bar-recursion of Spector [Reference Spector and Dekker58], here extended to all types from $T^{\Omega ,S}$ (similar to, e.g., [Reference Kohlenbach26]).

This result extends to any suitable extension of the language of $\mathcal {F}^\omega $ (e.g., by any kind of new types and constants) together with any number of additional universal axioms in that language.

We omit the proof as it is almost exactly the same as the proof given for the analogous soundness result in [Reference Kohlenbach25] (although this result from [Reference Kohlenbach25] is of course not formulated for the system $\mathcal {F}^\omega $ ).

Besides the soundness of the Dialectica interpretation (together with the negative translation), the other one of the two central tools utilized in the metatheorems is that of majorizability. Originally introduced by Howard [Reference Howard and Troelstra19], the notion of majorizable functionals was later extended by Bezem [Reference Bezem5] to that of strongly majorizable functionals to provide a model for finite type arithmetic extended by the schema of bar recursion discussed before. In that way, this model of strongly majorizable functionals provides the crucial basis for proof mining metatheorems of systems allowing for dependent choice. In the context of the abstract types, we further need to consider an extension of this notion of strongly majorizable functionals to these new types. The first such extensions to abstract types have been devised in [Reference Gerhardy and Kohlenbach12, Reference Kohlenbach25]. However, in this setting (and essentially in all other settings for metatheorems proved afterwards), this extension was motivated and based on the metric structure assumed for the respective classes of spaces that were treated. We thus find ourselves here at a “fork in the road,” where we have to extend the notion of majorizability sensibly to our types $\Omega $ and S, both representing spaces which do not carry any metric structure. The key insight, already mentioned and motivated throughout the previous sections many times (e.g., in the context of the admissibility of the axioms containing existential quantifiers over variables of types $\Omega $ , S or $S(0)$ , etc.) is to

1. 1. majorize objects $A^S$ by natural numbers bounding the measure of A,

2. 2. majorize objects $x^\Omega $ by natural numbers bounding the measure of the full set $\Omega $ in S.

In the case of a probability measure, any object of type $\Omega $ or S is therefore uniformly majorized by $1$ but with the phrasing of (1) and (2), we wanted to highlight the general idea of this approach as it might be feasible also for more general finite contents.

In any way, similar to [Reference Gerhardy and Kohlenbach12, Reference Kohlenbach25], the majorants for objects with types from $T^{\Omega ,S}$ are objects with types from T according to the following extended projection:

Definition 8.4 (essentially [Reference Gerhardy and Kohlenbach12]).

Define $\widehat {\tau }\in T$ , given $\tau \in T^{\Omega ,S}$ , by recursion on the structure via

$$\begin{align*}\widehat{0}:=0,\;\widehat{\Omega}:=0,\;\widehat{S}:=0,\;\widehat{\tau(\xi)}:=\widehat{\tau}(\widehat{\xi}). \end{align*}$$

The majorizability relation $\gtrsim $ is then defined in tandem with the structure of all strongly majorizable functionals.

Definition 8.5 (essentially [Reference Gerhardy and Kohlenbach12, Reference Kohlenbach25]).

Let $\Omega $ be a nonempty set, $S\subseteq 2^\Omega $ be an algebra and $\mathbb {P}$ be a probability content on S. The structure $\mathcal {M}^{\omega ,\Omega ,S}$ and the majorizability relation $\gtrsim _\rho $ are defined by

$$\begin{align*}\begin{cases} \mathcal{M}_0:=\mathbb{N}, n\gtrsim_0 m:=n\geq m\land n,m\in\mathbb{N},\\ \mathcal{M}_\Omega:= \Omega, n\gtrsim_\Omega x:= n\geq \mathbb{P}(\Omega)\land n\in \mathcal{M}_0,x\in \mathcal{M}_\Omega,\\ \mathcal{M}_{S}:= S, n\gtrsim_{S} A:= n\geq \mathbb{P}(A)\land n\in \mathcal{M}_0,A\in \mathcal{M}_{S},\\ f\gtrsim_{\tau(\xi)}x:=f\in \mathcal{M}_{\widehat{\tau}}^{\mathcal{M}_{\widehat{\xi}}}\land x\in \mathcal{M}_\tau^{\mathcal{M}_\xi}\\ \phantom{f\gtrsim_{\tau(\xi)}x:=}\land\forall g\in \mathcal{M}_{\widehat{\xi}},y\in \mathcal{M}_\xi(g\gtrsim_\xi y\rightarrow fg\gtrsim_\tau xy)\\ \phantom{f\gtrsim_{\tau(\xi)}x:=}\land\forall g,y\in \mathcal{M}_{\widehat{\xi}}(g\gtrsim_{\widehat{\xi}}y\rightarrow fg\gtrsim_{\widehat{\tau}}fy),\\ \mathcal{M}_{\tau(\xi)}:=\left\{x\in \mathcal{M}_\tau^{\mathcal{M}_\xi}\mid \exists f\in \mathcal{M}^{\mathcal{M}_{\widehat{\xi}}}_{\widehat{\tau}}:f\gtrsim_{\tau(\xi)}x\right\}. \end{cases} \end{align*}$$

So, as already discussed previously, though only informally, as $\mathbb {P}$ is a probability content, we have $1\gtrsim _S A$ for any $A\in S$ (as $\mathbb {P}(A)\leq \mathbb {P}(\Omega )=1$ ) as well as $1\gtrsim _\Omega x$ for any $x\in \Omega $ (but in the way the model is defined above, the definition immediately makes sense in the context of general finite contents).

Before we move on further, we now just quickly note that majorizability behaves nicely w.r.t. functions with multiple arguments as represented by their curried variants.

The other main structure featuring in the metatheorems is the structure of all set-theoretic functionals $\mathcal {S}^{\omega ,\Omega ,S}$ , defined via $\mathcal {S}_0:=\mathbb {N}$ , $\mathcal {S}_\Omega := \Omega $ , $\mathcal {S}_{S}:=S$ and

$$\begin{align*}\mathcal{S}_{\tau(\xi)}:=\mathcal{S}_{\tau}^{\mathcal{S}_{\xi}}. \end{align*}$$

Both structures $\mathcal {S}^{\omega ,\Omega ,S}$ and $\mathcal {M}^{\omega ,\Omega ,S}$ later turn into models of our systems if equipped with corresponding interpretations for the respective additional constants, with $\mathcal {S}^{\omega ,\Omega ,S}$ serving as the structure for the intended standard models.

The proof of the bound extraction theorems now follows the following general high-level outline of most other such metatheorems: using functional interpretation and negative translation, one extracts realizers from (essentially) $\forall \exists $ -theorems. These realizers have types from $T^{\Omega ,S}$ . We then use majorizability to construct bounds for these realizers, depending only on majorants of the parameters, which are validated in a model based on $\mathcal {M}^{\omega ,\Omega ,S}$ . In a final step, we can then recover to the truth in a model based on the usual full set-theoretic structure $\mathcal {S}^{\omega ,\Omega ,S}$ if the types occurring in the axioms and the theorem are “low enough,” which we will call admissible. Concretely, following [Reference Gerhardy and Kohlenbach12, Reference Kohlenbach25], we introduce the following specific classes of types: We call a type $\xi $ of degree n if $\xi \in T$ and it has degree $\leq n$ in the usual sense (see, e.g., [Reference Kohlenbach26]). Further we call $\xi $ small if it is of the form $\xi =\xi _0(0)\dots (0)$ for $\xi _0\in \{0,\Omega ,S\}$ (including $0,\Omega ,S$ ) and call it admissible if it is of the form $\xi =\xi _0(\tau _k)\dots (\tau _1)$ where each $\tau _i$ is small and $\xi _0\in \{0,\Omega ,S\}$ (also including $0,\Omega ,S$ ).

Further, also in analogy to [Reference Gerhardy and Kohlenbach12, Reference Kohlenbach25], we define certain subclasses of formulas satisfying certain type restrictions: A formula A is called a $\forall $ -formula if $A=\forall \underline {a}^{\underline {\xi }} A_{qf}(\underline {a})$ with $A_{qf}$ quantifier-free and all types $\xi _i$ in $\underline {\xi }=(\xi _1,\dots ,\xi _k)$ are admissible. A formula A is called an $\exists $ -formula if $A=\exists \underline {a}^{\underline {\xi }}A_{qf}(\underline {a})$ with similar $A_{qf}$ and $\underline {\xi }$ .

The class $\Delta $ already mentioned previously, which was originally introduced in [Reference Kohlenbach20, Reference Kohlenbach21] (and then lifted to abstract types in [Reference Günzel and Kohlenbach16]) to signify a collection of commonly occurring formulas with trivial monotone functional interpretations, is now similarly introduced in the context of the systems studied in this paper: a formula of type $\Delta $ is any formula of the form

$$\begin{align*}\forall\underline{a}^{\underline{\delta}}\exists\underline{b}\leq_{\underline{\sigma}}\underline{r}\underline{a}\forall\underline{c}^{\underline{\gamma}}A_{qf}(\underline{a},\underline{b},\underline{c}) \end{align*}$$

where $A_{qf}$ is quantifier-free, the types in $\underline {\delta }$ , $\underline {\sigma }$ , and $\underline {\gamma }$ are admissible, $\underline {r}$ is a tuple of closed terms of appropriate type, $\leq $ is defined by recursion on the type via

1. 1. $x\leq _0 y:=x\leq _0 y$ ,

2. 2. $x\leq _\Omega y:=\mathbb {P}(\Omega )\leq _{\mathbb {R}}\mathbb {P}(\Omega )$ ,

3. 3. $A\leq _{S} B:=\mathbb {P}(A)\leq _{\mathbb {R}}\mathbb {P}(B)$ ,

4. 4. $x\leq _{\tau (\xi )} y:=\forall z^\xi (xz\leq _\tau yz)$ ,

and $\underline {x}\leq _{\underline {\sigma }}\underline {y}$ is an abbreviation for $x_1\leq _{\sigma _1}y_1\land \dots \land x_k\leq _{\sigma _k}y_k$ where $\underline {x}$ , $\underline {y}$ , and $\underline {\sigma }$ are k-tuples of terms and types, respectively, such that $x_i$ and $y_i$ are of type $\sigma _i$ .

Given a set $\beth $ of formulas of type $\Delta $ , we write $\widetilde {\beth }$ for the set of all Skolem normal forms

$$\begin{align*}\exists\underline{B}\leq_{\underline{\sigma}(\underline{\delta})}\underline{r}\forall\underline{a}^{\underline{\delta}}\forall\underline{c}^{\underline{\gamma}}A_{qf}(\underline{a},\underline{B}\underline{a},\underline{c}) \end{align*}$$

for any $\forall \underline {a}^{\underline {\delta }}\exists \underline {b}\leq _{\underline {\sigma }}\underline {r}\underline {a}\forall \underline {c}^{\underline {\gamma }}A_{qf}(\underline {a},\underline {b},\underline {c})$ in $\beth $ .

Remark 8.7. We want to note briefly that all axioms that were previously discussed as admissible based on our extended notion of majorizability can actually be seen as statements of type $\Delta $ in the context of the above definition. At first, the axiom $(\mathbb {P})_4$ can be equivalently rewritten as

$$\begin{align*}\forall A^S,B^S\exists x^\Omega\leq_\Omega c_\Omega(\mathbb{P}(A)>_{\mathbb{R}}\mathbb{P}(B)\to \left( x\in A \land x\not\in B\right)) \end{align*}$$

and is thus immediately of type $\Delta $ .Footnote ²⁰ Second, also the axiom $(I)_3$ can be rewritten with the additional boundedness information via

$$ \begin{align*} &\forall f^{1(\Omega)},a^1,b^1\exists A^S\leq_S \Omega\forall x^\Omega\left(f\in I\to \left(x\in f^{-1}([a,b])\leftrightarrow x\in A\right)\right) \end{align*} $$

and thus is of type $\Delta $ . Similarly, also the axiom $(I)_5$ can be rewritten as an axiom of type $\Delta $ as

$$\begin{align*}\forall f^{1(\Omega)},k^0\exists x^\Omega\leq_\Omega c_\Omega\left( f\in I\to \left( \left\lVert f\right\rVert_\infty-2^{-k}\leq_{\mathbb{R}} \vert f(x)\vert\right)\right). \end{align*}$$

Lastly, also the integrability axioms ${({\int })}_4$ and ${({\int })}_3$ can be rewritten as

$$\begin{align*}\forall f^{1(\Omega)}, k^0\exists x^\Omega\leq_\Omega c_\Omega\left( f\in I\land \int{f}\,\mathrm{d}\mathbb{P}<_{\mathbb{R}} -2^{-k}\to \left( f(x)\leq_{\mathbb{R}} -2^{-k}\right)\right) \end{align*}$$

and

$$ \begin{align*} \forall f^{1(\Omega)}, k^0\exists A^{S(0)}\leq_{S(0)}\lambda n^0.\Omega\left(f\in I\to \int{\left\vert f-\sum_{i=0}^{2^{k+1}b_f-1}\left(-b_f+\frac{i}{2^{k}}\right)\chi_{A(i)}\right\vert}\,\mathrm{d}\mathbb{P}\leq_{\mathbb{R}} 2^{-k}\right), \end{align*} $$

respectively, with $b_f:=\left \lVert f\right \rVert _\infty (0)+1$ as before, which turns them into axioms of type $\Delta $ .

Crucially, axioms of type $\Delta $ are trivialized under the monotone functional interpretationFootnote ²¹ and we treat any axiom of type $\Delta $ in $\mathcal {C}^\omega $ (or any suitable extension) “in this spirit.” We here only write “in this spirit” as we actually do not use a monotone variant of the Dialectica interpretation but treat the functional interpretation part and the majorization part of the combined interpretation separately. In that way, we follow the approach given in [Reference Günzel and Kohlenbach16] (see also the recent [Reference Pischke51]) and treat axioms of type $\Delta $ by employing a construction that converts a theory with axioms of such a type into a theory using only additional purely universal axioms formulated using the Skolem functions of these axioms. This new theory is then used in combination with the functional interpretation to extract the respective terms and then the proof proceeds as outlined before.

Concretely, we now proceed as follows: Let $\beth $ be a set of axioms of type $\Delta $ and write $\widehat {\mathcal {C}}^\omega $ for $\mathcal {C}^\omega $ without any of its axioms of type $\Delta $ . Then, we form a new theory $\overline {\mathcal {C}}^\omega _\beth $ from $\widehat {\mathcal {C}}^\omega $ by adding the Skolem functionals $\underline {B}$ of any axiom of type $\Delta $ of $\mathcal {C}^\omega +\beth $ , say of the form

$$\begin{align*}\forall\underline{a}^{\underline{\delta}}\exists\underline{b}\leq_{\underline{\sigma}}\underline{r}\underline{a}\forall\underline{c}^{\underline{\gamma}}A_{qf}(\underline{a},\underline{b},\underline{c}), \end{align*}$$

as new constants to the language and simultaneously adding the corresponding “instantiated Skolem normal form,” that is,

$$\begin{align*}\underline{B}\leq_{\underline{\sigma}(\underline{\delta})}\underline{r}\land \forall\underline{a}^{\underline{\delta}}\forall\underline{c}^{\underline{\gamma}}A_{qf}(\underline{a},\underline{B}\underline{a},\underline{c}), \end{align*}$$

as a new axiom. Therefore, the system $\overline {\mathcal {C}}^\omega _\beth $ only extends $\mathcal {F}^\omega $ by new types, constants, and universal axioms. Therefore, as mentioned before, Lemma 8.3 also applies to this system where the conclusion is then proved in $\overline {\mathcal {C}}^{\omega -}_\beth +(\mathrm {BR})$ , that is, $\overline {\mathcal {C}}^{\omega }_\beth $ with the principles $\textsf {QF}\mbox {-}\textsf {AC}$ and $\textsf {DC}$ removed and where the scheme of simultaneous bar-recursion is added.

In the case where the extension $\mathcal {C}^\omega $ contains the rule ${({\bigcup })}_2$ , we for simplicity assume that in the process of forming the extended theory, this rule is also removed in the sense that $\overline {\mathcal {C}}^{\omega }_\beth $ does not contain the rule and for any provable premise

$$\begin{align*}\mathcal{C}^\omega+\beth\vdash F_{qf}\to \forall n^0\left( A(n)\subseteq_SB\right), \end{align*}$$

we add the corresponding conclusion

$$\begin{align*}F_{qf}\to \bigcup A\subseteq_S B \end{align*}$$

as an axiom of $\overline {\mathcal {C}}^{\omega }_\beth $ .

We now move on to the central result of the majorization part of the chosen approach to the bound extraction theorems, stating that every closed term in the underlying language of the system in question is majorizable. As such, the result is similar to Lemma 9.11 in [Reference Gerhardy and Kohlenbach12].

Lemma 8.8. Let $\mathcal {C}^\omega $ be (one of the previously discussed extensions of) the system $\mathcal {F}^\omega [\mathbb {P}]$ and let $\beth $ be a set of additional axioms of type $\Delta $ . Let $\Omega $ be a nonempty set and let $S\subseteq 2^\Omega $ be an algebra (or, in the context of the constant $\bigcup $ , a $\sigma $ -algebra). Let $\mathbb {P}$ be a probability content on S. Then $\mathcal {M}^{\omega ,\Omega ,S}$ is a model of $\overline {\mathcal {C}}^{\omega -}_\beth +(\mathrm {BR})$ , provided $\mathcal {S}^{\omega ,\Omega ,S}\models \beth $ (with $\mathcal {M}^{\omega ,\Omega ,S}$ and $\mathcal {S}^{\omega ,\Omega ,S}$ defined via suitable interpretations of the additional constants in $\mathcal {C}^\omega $ ). Moreover, for any closed term t of $\overline {\mathcal {C}}^{\omega -}_\beth +(\mathrm {BR})$ , one can construct a closed term $t^*$ of $\mathcal {A}^\omega +(\mathrm {BR})$ such that

$$\begin{align*}\mathcal{M}^{\omega,\Omega,S}\models\left( t^*\gtrsim t\right). \end{align*}$$

Proof. The structure of the proof is standard and similar to proofs of related results from the literature (see, e.g., [Reference Kohlenbach26]). As such, we only discuss the interpretations of the new constants added to $\mathcal {A}^\omega $ to form the respective theories as well as their majorizations. For the constants already contained in $\mathcal {A}^\omega $ , we may choose suitable interpretations as in [Reference Kohlenbach26] and for majorizing a composition of terms, we may similarly proceed as outlined therein. For that, we now first focus on $\mathcal {F}^\omega [\mathbb {P}]$ and assume that there are no further axioms of type $\Delta $ beyond those already contained in $\mathcal {F}^\omega [\mathbb {P}]$ . For any $\mathcal {C}^\omega $ , we deal with any set $\beth $ of additional axioms of type $\Delta $ and the respectively induced constants later on by moving to the theory $\overline {\mathcal {C}}^\omega _\beth $ .

Now, for the new constants added to $\mathcal {A}^\omega $ to form $\mathcal {F}^\omega [\mathbb {P}]$ , we consider the following interpretations (writing $\mathcal {M}$ for $\mathcal {M}^{\omega ,\Omega ,S}$ ):

• • $[\mathrm {eq}]_{\mathcal {M}}:=\text {the characteristic function of the equality relation in }\Omega $ ;

• • $[\in ]_{\mathcal {M}}:=\text {the characteristic function of the element relation in }S$ ;

• • $[\cup ]_{\mathcal {M}}:=\text {union in }S$ ;

• • $[(\cdot )^c]_{\mathcal {M}}:=\text {complement in }S$ ;

• • $[\emptyset ]_{\mathcal {M}}:=\text {the empty set in }S$ ;

• • $[\mathbb {P}]_{\mathcal {M}}:= \lambda A^S.(\mathbb {P}(A))_\circ $ where $\mathbb {P}$ is the content fixed in the context of this lemma.

This is only well-defined in $\mathcal {M}^{\omega ,\Omega ,S}$ if we can construct majorants of these objects. This we can do as follows:

• • $\lambda x^0,y^0.1\gtrsim \mathrm {eq}$ ;

• • $\lambda x^0,y^0.1\gtrsim \, \in $ ;

• • $\lambda x^0,y^0.1\gtrsim \cup $ ;

• • $\lambda x^0. 1\gtrsim (\cdot )^c$ ;

• • $0\gtrsim \emptyset $ ;

• • $\lambda x^0.(x)_\circ \gtrsim \mathbb {P}$ .

Note that in the last item, the operation $(x)_\circ $ is definable in $\mathcal {A}^\omega $ via a closed term as x is of type $0$ .

For justifying that those terms really are majorants of the respective constants, we argue as follows: The first four items immediately follow from the fact that $\mathbb {P}(A)\leq \mathbb {P}(\Omega )=1$ (i.e., that $\mathbb {P}$ is a probability content) and that $\mathbb {P}(\emptyset )=_{\mathbb {R}}0$ . The last item follows immediately from Lemma 2.1 as clearly, if $x\geq _{\mathbb {R}}\mathbb {P}(A)$ , then $(x)_\circ \gtrsim (\mathbb {P}(A))_\circ $ .

In the case where $\mathcal {C}^\omega $ contains the respective additional constant $\bigcup $ , a corresponding interpretation is naturally defined by

• • $[\bigcup ]_{\mathcal {M}}:=\text {countably infinite union in S}$ ,

which is well-defined since we in this context assume that S is a $\sigma $ -algebra. We can achieve majorization as before by exploiting that $\mathbb {P}$ is a finite content with

• • $\lambda f^{0(0)}.1\gtrsim \bigcup $ .

Lastly, if the system $\mathcal {C}^\omega $ contains the respective constants and axioms for treating integrals, we choose corresponding interpretations of the additional constants as follows:

• • $\big [[\cdot ,\cdot ]\big ]_{\mathcal {M}}:=\lambda a^1,b^1,x^1.\begin {cases}0&\text {if }r_x\in [r_a,r_b];\\1&\text {otherwise};\end {cases}$

• • $[(\cdot )^{-1}]_{\mathcal {M}}:=\lambda f^{1(\Omega )},A^{0(1)},x^\Omega .\begin {cases}0&\text {if }x\in f^{-1}(\{r_a\mid a^1: A(a)=_00\});\\1&\text {otherwise};\end {cases}$

• • $[I]_{\mathcal {M}}:=$ the characteristic function of a set of bounded and weakly Borel-measurable functions $f^{1(\Omega )}$ which is closed under linear combinations, multiplication with characteristic functions, and absolute values;

• • $[\left \lVert \cdot \right \rVert _\infty ]_{\mathcal {M}}:=\lambda f^{1(\Omega )}.\begin {cases}(\sup _{x\in \Omega }\vert f(x)\vert )_\circ &\text {if }f\text { is bounded};\\0&\text {otherwise};\end {cases}$

• • $[\int {\cdot }\,\mathrm {d}\mathbb {P}]_{\mathcal {M}}:=\lambda f^{1(\Omega )}.\begin {cases}(\int {f}\,\mathrm {d}\mathbb {P})_\circ &\text {if }f\text { is bounded and weakly Borel-measurable};\\0&\text {otherwise};\end {cases}$

  where the latter $\int {f}\,\mathrm {d}\mathbb {P}$ represents the usual integral defined over the content.

Note that here, we now rely on the extended operator $(\cdot )_\circ $ operating on all real numbers as the integral of a general integrable function may be negative. As for majorization, we rely on the following constructions:

• • $\lambda a^1,b^1,r^1.1\gtrsim [\cdot ,\cdot ]$ ;

• • $\lambda f^{1(0)},A^{0(1)},x^0.1\gtrsim (\cdot )^{-1}$ ;

• • $\lambda f^{1(0)}.1\gtrsim I$ ;

• • $\lambda f^{1(0)}.(f(1)(0)+1)_\circ \gtrsim \left \lVert \cdot \right \rVert _\infty $ ;

• • $\lambda f^{1(0)}.(f(1)(0)+1)_\circ \gtrsim \int {\cdot }\,\mathrm {d}\mathbb {P}$ .

The first three items are immediate as we deal with characteristic functions. For the fourth, note that if ${f^*}^{1(0)}\gtrsim f^{1(\Omega )}$ , then

$$\begin{align*}\forall n^0,x^\Omega\left( n\gtrsim x\to f^*(n)\gtrsim f(x)\right) \end{align*}$$

and as $1\gtrsim x$ for any $x^\Omega $ as $1=\mathbb {P}(\Omega )$ , we have $f^*(1)\gtrsim f(x)$ for any $x^\Omega $ . Therefore, we have

$$\begin{align*}f^*(1)(0)+1\geq_{\mathbb{R}}f(x)(0)+1\geq_{\mathbb{R}}[\vert f(x)\vert](0)+1\geq_{\mathbb{R}} \vert f(x)\vert \end{align*}$$

for any $x^\Omega $ , by the monotonicity of the coding of rational numbers (again, see, e.g., the discussion on p.430 in [Reference Kohlenbach26]). This implies $f^*(1)(0)+1\geq _{\mathbb {R}} \left \lVert f\right \rVert _\infty $ so that the result follows from Lemma 2.1.

Lastly, note that for any bounded and weakly Borel-measurable function f, we have that $\vert \int {f}\,\mathrm {d}\mathbb {P}\vert \leq _{\mathbb {R}}\int {\vert f\vert }\,\mathrm {d}\mathbb {P}$ so that

$$\begin{align*}\left\vert \int{f}\,\mathrm{d}\mathbb{P}\right\vert\leq_{\mathbb{R}}\int{\vert f\vert}\,\mathrm{d}\mathbb{P}\leq_{\mathbb{R}} \left\lVert\vert f\vert\right\rVert_\infty=_{\mathbb{R}}\left\lVert f\right\rVert_\infty\leq_{\mathbb{R}} f^*(1)(0)+1 \end{align*}$$

as before. The majorizability result then follows again from Lemma 2.1.

That $\mathcal {M}^{\omega ,\Omega ,S}$ with these chosen interpretations is a model of $\mathcal {C}^{\omega -}+(\mathrm {BR})$ can be shown similarly to analogous results (see, e.g., [Reference Kohlenbach26]). The intended interpretations of the constants of $\mathcal {C}^\omega $ in $\mathcal {S}^{\omega ,\Omega ,S}$ , turning $\mathcal {S}^{\omega ,\Omega ,S}$ into a model of these systems, are defined in analogy to the corresponding model $\mathcal {M}^{\omega ,\Omega ,S}$ defined above.

For treating the other additional axioms in $\mathcal {C}^\omega +\beth $ of type $\Delta $ beyond the axioms already contained in $\mathcal {C}^\omega $ , we rely on the following argument (akin to [Reference Günzel and Kohlenbach16], Lemma 5.11) showing that $\mathcal {S}^{\omega ,\Omega ,S}\models \beth $ implies $\mathcal {M}^{\omega ,\Omega ,S}\models \widetilde {\beth }$ . For this, the proof given in [Reference Günzel and Kohlenbach16] for Lemma 5.11 carries over which we sketch here: While $\mathcal {M}^{\omega ,\Omega ,S}$ in general is not a model of the axiom of choice [Reference Kohlenbach22], one can show (similar to [Reference Kohlenbach22]) that $\mathcal {M}^{\omega ,\Omega ,S}\models \mathsf {b}\text {-}\mathsf {AC}_{\Omega ,S}$ where

$$\begin{align*}\mathsf{b}\text{-}\mathsf{AC}_{\Omega,S}:=\bigcup_{\delta,\rho\in T^{\Omega,S}}\mathsf{b}\text{-}\mathsf{AC}^{\delta,\rho} \end{align*}$$

with

$$\begin{align*}\mathsf{b}\text{-}\mathsf{AC}^{\delta,\rho}:=\forall Z^{\rho(\delta)}\left( \forall x^\delta\exists y\leq_\rho Zx A(x,y,Z)\to \exists Y\leq_{\rho(\delta)} Z\forall x^\delta A(x,Yx,Z)\right). \end{align*}$$

Now, for small types $\rho $ , we have $M_\rho =S_\rho $ while for admissible types $\rho $ , we have $M_\rho \subseteq S_\rho $ (for which it is important that admissible types take arguments of small types). For this, the proof given in [Reference Gerhardy and Kohlenbach12] carries over. Further, we need that it is provable in $\mathcal {C}^{\omega -}$ that

(†) $$ \begin{align} \forall x',x,y\left( x'\gtrsim_\rho x\land x\geq_\rho y\to x'\gtrsim_\rho y\right) \end{align} $$

holds for all types $\rho $ which can be shown similar to, for example, [Reference Kohlenbach26].

Suppose now that

$$\begin{align*}\mathcal{S}^{\omega,\Omega,S}\models \forall\underline{a}^{\underline{\delta}}\exists\underline{b}\leq_{\underline{\sigma}}\underline{r}\underline{a}\forall\underline{c}^{\underline{\gamma}}A_{qf}(\underline{a},\underline{b},\underline{c}). \end{align*}$$

Then also $\mathcal {M}^{\omega ,\Omega ,S}$ is a model of this sentence: First the types of the variables which are universally quantified are admissible, so over $\mathcal {M}^{\omega ,\Omega ,S}$ the domain of the universal quantifiers is reduced. For the witnesses for $\underline {b}$ , which exist in $\mathcal {S}^{\omega ,\Omega ,S}$ , note first that these could potentially live in $\mathcal {M}^{\omega ,\Omega ,S}$ as the types of the variables in $\underline {b}$ are admissible, that is, they take arguments of small types and map into small types. It thus only remains to be seen whether such a witness is majorizable for majorizable inputs. However, by the above argument, the terms in $\underline {r}$ are all majorizable and if $\underline {a}$ comes from $\mathcal {M}^{\omega ,\Omega ,S}$ , then $\underline {r}\underline {a}$ is majorizable. That we have $\underline {b}\leq _{\underline {\sigma }}\underline {r}\underline {a}$ in $\mathcal {M}^{\omega ,\Omega ,S}$ now implies that $\underline {b}$ is majorizable by $(\dagger )$ (and consequently the corresponding interpretations exist in $\mathcal {M}^{\omega ,\Omega ,S}$ too). Lastly, it is rather immediate to see that $\mathcal {M}^{\omega ,\Omega ,S}\models \beth $ implies $\mathcal {M}^{\omega ,\Omega ,S}\models \widetilde {\beth }$ using $\mathsf {b}\text {-}\mathsf {AC}_{\Omega ,S}$ .

From $\mathcal {M}^{\omega ,\Omega ,S}\models \widetilde {\beth }$ , we immediately get that the corresponding Skolem functions have interpretations in $\mathcal {M}^{\omega ,\Omega ,S}$ , that the corresponding structures defined by some canonical interpretations of those additional constants are indeed models of those variants of the systems where the corresponding Skolem functionals of these axioms are added and where the axioms themselves are replaced by their instantiated Skolem normal forms (i.e., $\overline {\mathcal {C}}^{\omega -}_\beth $ and its extensions) and, lastly, that the above majorizability result extends to these systems.

Note that, technically, these arguments were already needed in the above considerations to see that $\mathcal {M}^{\omega ,\Omega ,S}$ really is a model of $\mathcal {C}^{\omega -}+(\mathrm {BR})$ . However, we did not discuss this there explicitly as for those specific axioms of type $\Delta $ belonging to $\mathcal {C}^{\omega -}+(\mathrm {BR})$ , the types of the variables occurring in them are not only small but actually all among $\{0,1,\Omega ,S,S(0)\}$ so that it was immediately clear that the models coincide at that level (essentially just by definition) and we thus omitted such a general discussion there.

Combined with the Dialectica interpretation, the main result we then arrive at is the following bound extraction result for classical proofs:

Theorem 8.9. Let $\mathcal {C}^\omega $ be (one of the previously discussed extensions of) the system $\mathcal {F}^\omega [\mathbb {P}]$ and let $\beth $ be a set of formulas of type $\Delta $ . Let $\tau $ be admissible, $\delta $ be of degree $1$ and s be a closed term of $\mathcal {C}^\omega $ of type $\sigma (\delta )$ for admissible $\sigma $ and let $B_\forall (x,y,z,u)$ / $C_\exists (x,y,z,v)$ be $\forall $ -/ $\exists $ -formulas of $\mathcal {C}^{\omega }$ with only $x,y,z,u$ / $x,y,z,v$ free. If

$$\begin{align*}\mathcal{C}^{\omega}+\beth\vdash\forall x^\delta\forall y\leq_\sigma s(x)\forall z^\tau\left(\forall u^0 B_\forall(x,y,z,u)\rightarrow\exists v^0 C_\exists(x,y,z,v)\right), \end{align*}$$

then one can extract a partial functional $\Phi :\mathcal {S}_{\delta }\times \mathcal {S}_{\widehat {\tau }}\rightharpoonup \mathbb {N}$ which is total and (bar-recursively) computable on $\mathcal {M}_\delta \times \mathcal {M}_{\widehat {\tau }}$ and such that for all $x\in \mathcal {S}_\delta $ , $z\in \mathcal {S}_\tau $ , $z^*\in \mathcal {S}_{\widehat {\tau }}$ , if $z^*\gtrsim z$ , then

$$\begin{align*}\mathcal{S}^{\omega,\Omega,S}\models\forall y\leq_\sigma s(x)\left(\forall u\leq_0\Phi(x,z^*) B_\forall(x,y,z,u)\rightarrow\exists v\leq_0\Phi(x,z^*)C_\exists(x,y,z,v)\right) \end{align*}$$

holds whenever $\mathcal {S}^{\omega ,\Omega ,S}\models \beth $ for $\mathcal {S}^{\omega ,\Omega ,S}$ defined via any nonempty set $\Omega $ and any algebra $S\subseteq 2^\Omega $ (or, in the context of the constant $\bigcup $ , any $\sigma $ -algebra) together with any probability content $\mathbb {P}$ on S (and with suitable interpretations of the additional constants). Further:

1. 1. If $\widehat {\tau }$ is of degree $1$ , then $\Phi $ is a total computable functional.

2. 2. We may have tuples instead of single variables $x,y,z,u,v$ and a finite conjunction instead of a single premise $\forall u^0 B_\forall (x,y,z,u)$ .

3. 3. If the claim is proved without $\textsf {DC}$ , then $\tau $ may be arbitrary and $\Phi $ will be a total functional on $\mathcal {S}_\delta \times \mathcal {S}_{\widehat {\tau }}$ which is primitive recursive in the sense of Gödel [Reference Gödel13] and Hilbert [Reference Hilbert18]. In that case, also plain majorization can be used instead of strong majorization (see, e.g., [Reference Kohlenbach26]).

Proof. First, assume that $\mathcal {S}^{\omega ,\Omega ,S}\models \beth $ and (for simplicity) that

$$\begin{align*}\mathcal{C}^{\omega}+\beth\vdash\forall z^\tau\left(\forall u^0 B_\forall(z,u)\rightarrow\exists v^0 C_\exists(z,v)\right). \end{align*}$$

Clearly, the same statement is then also provable in $\overline {\mathcal {C}}^\omega _\beth $ . By assumption, $B_\forall (z,u)=\forall \underline {a} B_{qf}(z,u,\underline {a})$ and $C_\exists (z,v)=\exists \underline {b} C_{qf}(z,v,\underline {b})$ for quantifier-free $B_{qf}$ and $C_{qf}$ . Thus, by prenexiation, we get

$$\begin{align*}\overline{\mathcal{C}}^\omega_\beth\vdash\forall z^\tau\exists u,v,\underline{a},\underline{b}(B_{qf}(z,u,\underline{a})\rightarrow C_{qf}(z,v,\underline{b})). \end{align*}$$

Using Lemma 8.3 (which is applicable as $\overline {\mathcal {C}}^\omega _\beth $ is an extension of $\mathcal {F}^\omega $ only by new constants and purely universal axioms) and disregarding the realizers for $\underline {a},\underline {b}$ , we get closed terms $t_u,t_v$ of $\overline {\mathcal {C}}^{\omega -}_\beth +(\mathrm {BR})$ such that

$$\begin{align*}\overline{\mathcal{C}}^{\omega-}_\beth+(\mathrm{BR})\vdash\forall z^\tau(B_\forall (z,t_u(z))\rightarrow C_\exists(z,t_v(z))). \end{align*}$$

By Lemma 8.8 there are closed terms $t^*_u,t^*_v$ of $\mathcal {A}^\omega +(\mathrm {BR})$ such that

$$\begin{align*}\mathcal{M}^{\omega,\Omega,S}\models t^*_u\gtrsim t_u\land t^*_v\gtrsim t_v\land\forall z^\tau(B_\forall(z,t_u(z))\rightarrow C_\exists(z,t_v(z))) \end{align*}$$

for all nonempty sets $\Omega $ , any algebra (or $\sigma $ -algebra) $S\subseteq 2^\Omega $ and any probability content $\mathbb {P}$ on S and where the constants are interpreted as in Lemma 8.8. Define

$$\begin{align*}\Phi(z^*):=\max\{t^*_u(z^*),t^*_v(z^*)\}. \end{align*}$$

Then

$$\begin{align*}\mathcal{M}^{\omega,\Omega,S}\models\forall u\leq_0\Phi(z^*) B_\forall(z,u)\rightarrow\exists v\leq_0\Phi(z^*) C_\exists(z,v) \end{align*}$$

holds for all $z\in \mathcal {M}_\tau $ and $z^*\in \mathcal {M}_{\widehat {\tau }}$ with $z^*\gtrsim z$ . The conclusion that $\mathcal {S}^{\omega ,\Omega ,S}$ satisfies the same sentence can be achieved as in the proof of Theorem 17.52 in [Reference Kohlenbach26] which we sketch here: Note that in the conclusion, we restrict ourselves to those z which have majorants $z^*$ . As the type of z is admissible, it takes arguments of small type for which $\mathcal {M}^{\omega ,\Omega ,S}$ and $\mathcal {S}^{\omega ,\Omega ,S}$ coincide (recall the proof of Lemma 8.8). Therefore, any such $z,z^*$ from $\mathcal {S}^{\omega ,\Omega ,S}$ also live in $\mathcal {M}^{\omega ,\Omega ,S}$ so that $\Phi (z^*)$ is well-defined for $z,z^*$ belonging to $\mathcal {S}^{\omega ,\Omega ,S}$ with $z^*\gtrsim z$ . In $B_\forall $ , all types are admissible so that truth in $\mathcal {S}^{\omega ,\Omega ,S}$ implies truth in $\mathcal {M}^{\omega ,\Omega ,S}$ and similarly for $C_\exists $ where thus truth in $\mathcal {M}^{\omega ,\Omega ,S}$ implies truth in $\mathcal {S}^{\omega ,\Omega ,S}$ . Lastly, as in Lemma 17.84 in [Reference Kohlenbach26], we can show that as $\Phi $ is of type $0(\widehat {\tau })$ , the interpretations of $\Phi $ in $\mathcal {S}^{\omega ,\Omega ,S}$ and $\mathcal {M}^{\omega ,\Omega ,S}$ coincide on majorizable elements. All in all, the arguments above imply that

$$\begin{align*}\mathcal{S}^{\omega,\Omega,S}\models\forall u\leq_0\Phi(z^*) B_\forall(z,u)\rightarrow\exists v\leq_0\Phi(z^*) C_\exists(z,v) \end{align*}$$

holds for all $z\in S_\tau $ and $z^*\in S_{\widehat {\tau }}$ with $z^*\gtrsim z$ .

The additional $\forall x^\delta \forall y\leq _\sigma s(x)$ can be treated as, for example, discussed in [Reference Kohlenbach25] and we thus omit any details. Similarly, item (1) can be shown as in the proof of Theorem 17.52 from [Reference Kohlenbach26] (see page 428 therein). Further, (2) is immediate and (3) follows from the fact that without $\textsf {DC}$ , bar recursion becomes superfluous, and the model $\mathcal {M}^{\omega ,\Omega ,S}$ can be avoided.

9 Applications of the metatheorems

In this section, we are now concerned with the applications of the above metatheorems. Concretely, we want to indicate how the systems introduced prior can be used together with their metatheorems to recognize previous (ad hoc) applications in the spirit of the proof mining program as proper instances of proof-theoretic bound extraction theorems. For this, we here focus on the seminal work [Reference Avigad, Dean and Rute2] by Avigad, Dean, and Rute. There, we find that the quantitative results obtained in [Reference Avigad, Dean and Rute2] for Egorov’s theorem, as well as the dominated convergence theorem, although in general being of bar-recursive complexity (which is due to the use of a certain principle of countable choice as we will later see formally), are nevertheless highly uniform, being in particular independent of the space, the measurable sets, and the measure. As already mentioned in the introduction, the authors of [Reference Avigad, Dean and Rute2] presumed that this independence can be explained using some instantiation of the notion of majorizability. In that way, the metatheorems proved before and the discussion in this section show that this intuition was correct and that the uniformities are a necessary consequence of the novel form of majorizability introduced in this paper.

However, as already discussed in the introduction, we want to mention that the focus on the work [Reference Avigad, Dean and Rute2] shall be understood to be merely indicative of the usefulness of the systems and metatheorems discussed before. In particular, essentially all other quantitative works on probability theory in the spirit of proof mining that have so far been considered in the literature can be similarly recognized as instances of the metatheorems proved here. Similarly, there too, the peculiar uniformities observed in practice are a priori guaranteed by the approach towards the metatheorems chosen here.

Now, the work [Reference Avigad, Dean and Rute2] is concretely concerned with interrelations between different modes of convergence for sequences of random variables. The most prominent of these modes, also based on its similarity to a usual notion of pointwise convergence of functions, is that of almost sure convergence.

Definition 9.1 (Almost sure convergence).

Let $(\Omega ,S,\mathbb {P})$ be a probability space and $(X_n)$ be a sequence of random variables $X_n:\Omega \to \mathbb {R}$ (i.e., $X_n$ is measurable w.r.t. S and the Borel $\sigma $ -algebra on $\mathbb {R}$ ). Then $(X_n)$ is said to converge almost surely to a random variable $X:\Omega \to \mathbb {R}$ if

$$\begin{align*}\mathbb{P}(\{x\in \Omega\mid X_n(x) \to X(x)\}) = 1. \end{align*}$$

This notion of almost sure convergence does not lend itself easily to a quantitative account of that convergence. Thus, in many cases where probability theorists are concerned with quantitative results (see, e.g., [Reference Luzia39, Reference Siegmund55]), they opt for a different, but equivalent, formulation of almost sure convergence known as almost uniform convergence.

Definition 9.2 (Almost uniform convergence).

Let $(\Omega ,S,\mathbb {P})$ be a probability space and $(X_n)$ be a sequence of random variables $X_n:\Omega \to \mathbb {R}$ . Then $(X_n)$ is said to converge almost uniformlyFootnote ²² to a random variable $X:\Omega \to \mathbb {R}$ if for all $\varepsilon , \delta>0$ there exists an $N \in \mathbb {N}$ such that

$$\begin{align*}\mathbb{P}(\{x\in \Omega\mid \forall n \ge N \left( |X_n(x) - X(x)| \le \varepsilon\right)\})> 1 - \delta. \end{align*}$$

The seminal result that these two notions of convergence are indeed equivalent is known as Egorov’s theorem [Reference Egoroff9]. Note that since $(\Omega ,S,\mathbb {P})$ is a probability space and the $X_n$ ’s are random variables (and therefore measurable), the sets we take the probability of in the above definitions are measurable sets.

This result was analyzed quantitatively in [Reference Avigad, Dean and Rute2] and then was used in turn also to provide a quantitative dominated convergence theorem for the Lebesgue integral. With the results from this section, we will be able to recognize this analysis as an instance of the preceding metatheorems for probability contents on algebras.

We now move towards formalizing the results from [Reference Avigad, Dean and Rute2] and for that purpose introduce a sequence of random variables into the system $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral}]$ by adding an additional constant $X^{1(\Omega )(0)}$ together with the axiom

$$\begin{align*}\forall n^0\left( X(n)\in I\right) \end{align*}$$

to stipulate that the sequence in question belongs to our subspace of bounded and weakly Borel-measurable functions. It is clear that Theorem 8.9 extends from $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral}]$ to its extension by this constant X as any $X(n)$ is trivially majorizable as it is bounded and the whole constant X is thus majorized by a maximum construction. In that system, using axiom $(I)_3$ that asserts the weak Borel-measurability of any $X(n)$ , it is then in particular a consequence of $\Pi ^\Omega _1$ - $\mathsf {AC}$ (and actually of $\mathsf {b}\text {-}\mathsf {AC}_{\Omega ,S}$ by regarding quantification over the type S as bounded) that there exists a functional $P^{S(0)(0)(0)}$ such that

$$\begin{align*}\forall a^0,b^0,c^0,x^\Omega(x \in P(a,b,c) \leftrightarrow x\in (\vert X(c)-X(b)\vert)^{-1}([0,2^{-a}])). \end{align*}$$

We add such a P directly into the language of the system via a new constant of the appropriate type together with the above defining property as an axiom and denote the resulting system by $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral},X]$ .

Using this functional P, we can then formally introduce the alternative way of formulating almost uniform convergence using finite unions as (implicitly) introduced in [Reference Avigad, Dean and Rute2], which allows for both a natural metastable variant as well as to extend any discussion regarding this notion naturally to the context of contents:

Definition 9.3. We say that X converges almost uniformly with respect to finite unions if

$$\begin{align*}\forall k^0, a^0\exists b^0\forall c^0\left( \mathbb{P}\kern1.5pt\left(\bigcup_{i=b}^c\bigcup_{j=b}^c P(a,i,j)^c\right)\le_{\mathbb{R}} 2^{-k}\right). \end{align*}$$

It is rather immediately clear that this notion of almost uniform convergence w.r.t. finite unions is equivalent over probability spaces to the usual notion of almost uniform convergence. Further, we want to emphasize that this mode was not explicitly introduced in [Reference Avigad, Dean and Rute2] but rather implicitly as already mentioned above as it is naturally suggested through the quantitative rendering of almost uniform convergence used in [Reference Avigad, Dean and Rute2] in their main quantitative result given in Theorem 3.1. Concretely, one immediately finds that a solution of the monotone functional interpretation of the negative translation of almost uniform convergence w.r.t. finite unions is exactly a function $M(k,a)$ providing a $2^{-k}$ -uniform bound for the $2^{-a}$ -metastable convergence of the sequence coded by X as introduced in [Reference Avigad, Dean and Rute2].

Similarly, we can also give a formal representation of the notion of almost uniform metastable pointwise convergence as introduced in [Reference Avigad, Dean and Rute2] in the context of the system $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral},X]$ :

Definition 9.4. We say that X converges almost uniform metastable pointwisely if

$$\begin{align*}\forall k^0,a^0,F^1\exists b^0\left( \mathbb{P}\kern1.5pt\left(\bigcap_{m=0}^b\bigcup_{i=m}^{F(m)}\bigcup_{j=m}^{F(m)}P(a,i,j)^c\right)\le_{\mathbb{R}} 2^{-k}\right). \end{align*}$$

Contrary to Definition 9.3, this mode of convergence was explicitly introduced in [Reference Avigad, Dean and Rute2] as a “more quantitatively friendly” version of the notion of almost sure convergence. In particular, as shown by Proposition 4.1 in [Reference Avigad, Dean and Rute2], the notion of almost uniform metastable pointwise convergence coincides over probability spaces with that of almost sure convergence. Also, as essentially already observed in [Reference Avigad, Dean and Rute2], a solution to the monotone functional interpretation (of the negative translation of) the statement of almost uniform metastable pointwise convergence is exactly a function $M'(k,a)$ providing a $2^{-k}$ -uniform bound on the $2^{-a}$ -metastable pointwise convergence of the sequence coded by X as introduced in [Reference Avigad, Dean and Rute2].

In [Reference Avigad, Dean and Rute2], the authors now provide a quantitative version of Egorov’s theorem by constructing a solution of the monotone functional interpretation of (the negative translation of) almost uniform convergence w.r.t. finite unions from a solution of the monotone functional interpretation of (the negative translation of) the statement of almost uniform metastable pointwise convergence. In that way, they in particular provide a uniform quantitative rendering of the corresponding implication. We justify this application by showing in the following that already the system $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral},X]$ proves this implication between the above two modes of convergence. Besides thereby explaining the success and the uniformities of the quantitative version of Egorov’s theorem from [Reference Avigad, Dean and Rute2], the provability in $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral},X]$ established here shows in particular that the corresponding results from [Reference Avigad, Dean and Rute2] are true for probability contents and not just probability measures as illustrated in [Reference Avigad, Dean and Rute2]. That is, the authors inadvertently provided an “Egorov-like theorem” for probability contents. Concretely, this seems to be in particular due to the above renderings of the notions of almost sure and almost uniform convergence introduced via a finitary perspective informed by proof mining in [Reference Avigad, Dean and Rute2], which provide exactly those alternative phrasings of these notions that are much more nicely compatible with the notion of a content due to a computationally effective formulation using finite unions. In that way, the results from this section tie into the comments made in the introduction that the notions and proofs produced through the finitary perspective of proof mining seem to be suitable so that they allow for a simultaneous lift of the results to the theory of contents.

We now first note that one direction of that equivalence can be easily witnessed in the system $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral},X]$ discussed previously.

Proof. We reason in $\mathcal {F}^\omega [\mathbb {P},\mathrm {Integral},X]$ . Let k, a, and F be given. Using that X converges almost uniformly w.r.t. finite unions, there exists a b such that

$$\begin{align*}\mathbb{P}\kern1.5pt\left(\bigcup_{i=b}^c\bigcup_{j=b}^c P(a,i,j)^c\right)\le 2^{-k} \end{align*}$$

for all c. Now, we in particular have

$$\begin{align*}\bigcap_{m=0}^b\bigcup_{i=m}^{F(m)}\bigcup_{j=m}^{F(m)} P(a,i,j)^c\subseteq \bigcup_{i=b}^{F(b)}\bigcup_{j=b}^{F(b)} P(a,i,j)^c \end{align*}$$

and therefore

$$\begin{align*}\mathbb{P}\kern1.5pt\left(\bigcap_{m=0}^b\bigcup_{i=m}^{F(m)}\bigcup_{j=m}^{F(m)} P(a,i,j)^c\right)\leq\mathbb{P}\kern1.5pt\left(\bigcup_{i=b}^{F(b)}\bigcup_{j=b}^{F(b)} P(a,i,j)^c\right)\leq 2^{-k} \end{align*}$$

follows from the monotonicity of $\mathbb {P}$ . This yields that X converges almost uniformly metastable pointwisely.

As mentioned before, a quantitative version of the converse of the above theorem is one of the main results of [Reference Avigad, Dean and Rute2] and to obtain this, the authors of [Reference Avigad, Dean and Rute2] mainly utilized a quantitative version of the following property about sequences of events.

Theorem 9.6 (Theorem 2.2 of [Reference Avigad, Dean and Rute2]).

For every sequence of events $(A_n)$ , any functional $M: \mathbb {N}^{\mathbb {N}}\to \mathbb {N}$ and any $\lambda> \lambda ' > 0$ , there exists an $n \in \mathbb {N}$ such that

$$\begin{align*}\mathbb{P}\kern1.5pt\left(\bigcap_{m = 0}^{M(F)}\bigcup_{j = m}^{F(m)}A_j\right) < \lambda'\text{ for all }F: \mathbb{N} \to \mathbb{N} \end{align*}$$

implies $\mathbb {P}(A_n) < \lambda $ .

We now discuss in the following how (the proof of) this result can be formalized in our system for probability contents on algebras $\mathcal {F}^\omega [\mathbb {P}]$ , justifying the existence and the uniformity of the quantitative result given in [Reference Avigad, Dean and Rute2] by means of our metatheorems. Concretely, we show:

Theorem 9.7. The system $\mathcal {F}^\omega [\mathbb {P}]$ proves:

$$\begin{align*}\forall A^{S(0)}, M^{0(1)}, u^0, v^0>_0 u\exists n^0\left(\forall F^1\left(\mathbb{P}\kern1.5pt\left(\bigcap_{m = 0}^{M(F)}\bigcup_{j = m}^{F(m)}A(j)\right) \le_{\mathbb{R}} 2^{-v} \right)\to\mathbb{P}(A(n)) <_{\mathbb{R}} 2^{-u}\right). \end{align*}$$

In particular, as this theorem of $\mathcal {F}^\omega [\mathbb {P}]$ has the correct logical form, our main Theorem 8.9 on the extraction of uniform computable bounds applies and we thus find that the existence of a computable bound on the existential quantifier on n can be guaranteed to exist a priori and even further, based on our notion of majorizability, it can be guaranteed that this bound is independent of the content space and the sequence of events which matches exactly the properties of the bound explicitly calculated in [Reference Avigad, Dean and Rute2].

To now demonstrate Theorem 9.7, we in particular rely on the following lemma:

Lemma 9.8. The system $\mathcal {F}^\omega [\mathbb {P}]$ proves:

$$\begin{align*}\forall A^{S(0)}, k^0\exists N^0\forall n^0\left(\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^n A(i) \cap \left(\bigcup_{i=0}^N A(i)\right)^c\right) <_{\mathbb{R}} 2^{-k} \right). \end{align*}$$

Proof. We reason in $\mathcal {F}^\omega [\mathbb {P}]$ . Let $A^{S(0)}$ and $k^0$ be given. At first, note that Proposition 4.5 implies that

(*) $$ \begin{align} \exists N \forall n \left(n \ge N \to \left\vert \sum_{i=0}^n\mathbb{P}((A\!\uparrow)(i)) - \sum_{i=0}^N\mathbb{P}((A\!\uparrow)(i))\right\vert< 2^{-k}\right). \end{align} $$

Take such an N and let n be arbitrary. If $n<N$ , then

$$\begin{align*}\bigcup_{i=0}^n A(i) \cap \left(\bigcup_{i=0}^N A(i)\right)^c=\emptyset \end{align*}$$

and so by extensionality of $\mathbb {P}$ , we get

$$\begin{align*}\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^n A(i) \cap \left(\bigcup_{i=0}^N A(i)\right)^c\right) = 0 \end{align*}$$

and are done. So suppose $n \ge N$ . Then by $(*)$ , we get

$$\begin{align*}\left\vert \sum_{i=0}^n\mathbb{P}((A\!\uparrow)(i)) - \sum_{i=0}^N\mathbb{P}((A\!\uparrow)(i))\right\vert < 2^{-k}. \end{align*}$$

Since all the $(A\!\uparrow )(i)$ are disjoint (by definition of $A\!\uparrow $ ) and since we have $\bigcup _{i=0}^j (A\!\uparrow )(i) = \bigcup _{i=0}^j A(i)$ for any j, we immediately derive

$$\begin{align*}\sum_{i=0}^j\mathbb{P}((A\!\uparrow)(i)) = \mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^j A(i) \right) \end{align*}$$

for any j by finite additivity and extensionality of $\mathbb {P}$ . Thus, we in particular have

$$\begin{align*}\left\vert \mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^n A(i) \right) - \mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^N A(i) \right)\right\vert< 2^{-k} \end{align*}$$

and since $n\geq N$ implies $\bigcup _{i=0}^N A(i) \subseteq \bigcup _{i=0}^n A(i)$ , we obtain

$$\begin{align*}\mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^n A(i) \cap \left(\bigcup_{i=0}^N A(i)\right)^c\right) = \mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^n A(i) \right) - \mathbb{P}\kern1.5pt\left(\bigcup_{i=0}^N A(i) \right)< 2^{-k} \end{align*}$$

by Proposition 4.3.