1. Introduction

Lewis proposed the Principal Principle in his seminal work (Reference Lewis1980). He argued that when a subjectivist believes in objective chance, the credences he assigns to events and the credences he assigns to possible values of the objective chances of these events cannot be arbitrary if the subjectivist is reasonable: these credences (apart from making a probability measure) have to satisfy a relation, namely, the relation that Lewis calls Principal Principle. Accordingly, in Lewis’s work, credence is assigned to at least two kinds of propositions: those expressing real events (e.g., the coin fell heads; we will call them “chancy events”) and those expressing that the objective chances of chancy events are equal to certain values (e.g., the chance of heads is, say, 35%). Recently, Rédei and Gyenis (Reference Rédei and Gyenis2013) have proposed at various forums, such as the PSA Biennial Meeting in 2014, that the Principal Principle makes sense only if whatever credence function is defined on the chancy events, such credence can be extended in a consistent manner to events concerning values of the objective chances of the chancy events. They presented their claim using the language of Bayesianism, but this is essentially what they proposed.

In this work I show that the extendibility Rédei and Gyenis require essentially always holds. Before doing so, however, I present an argument disputing the claim of Rédei and Gyenis that this extendibility is indeed necessary for the Principal Principle to be meaningful.

2. Credences of Events, Credences of Objective Chances

In this section we turn to Lewis (1980) for guidance and inspiration. We examine one of the examples he presented to motivate the Principal Principle, and we make some general observations. The example and the observations will guide us throughout the current work: On the one hand, they indicate how to construct the embedding Rédei and Gyenis require for their consistency. On the other hand, they help us understand the meaning of the Principal Principle, why it is a reasonable requirement, and why it would be a reasonable requirement even if Rédei and Gyenis’s consistency notion happened to fail to hold.

2.2. General Observations

Where does equation (1) come from? If A is some event of the real world, ${\mathrm{ch}}_t$ is the chance function over such events at some time t, and C is the credence function, the general idea that is expressed by the above example can be written as $C\left(A\right)={\sum }_{r}r\dot{}C\left({\mathrm{ch}}_t\right(A)=r)$, where the sum runs through all r values ${\mathrm{ch}}_t\left(A\right)$ can possibly take. Clearly, in case of a continuum, the summation has to be replaced by an integral. If we throw in Lewis’s admissible evidence too (think of it as facts from the past), then the above equation takes the following form:

(2)$\begin{array}{}C\left(A\right|E)=\sum _{r}r\dot{}C({\mathrm{ch}}_t\left(A\right)=r\left|E\right)\mathrm{.}\end{array}$

Note that if C satisfies the laws of probability, then the following statement always holds:

(3)$\begin{array}{}C\left(A\right|E)=\sum _{r}C(A\left|{\mathrm{ch}}_t\right(A)=r\wedge E)\dot{}C\left({\mathrm{ch}}_t\right(A)=r|E),\end{array}$

as long as ${\sum }_{r}C\left({\mathrm{ch}}_t\right(A)=r|E)=1$, that is, as long as r runs through all possible values of ${\mathrm{ch}}_t\left(A\right)$. The Principal Principle requires that

(4)$\begin{array}{}C\left(A\right|{\mathrm{ch}}_t\left(A\right)=r\wedge E)=r\end{array}$

hold for all admissible E events. Lewis (Reference Lewis1980) noted that equation (3) together with the Principal Principle (4) immediately implies equation (2).

The meaning of the Principal Principle is of course that a reasonable agent should choose the credence such that if the objective chance of a chancy event A is r according to the evidence collected by the agent, then the credence of A should be the same r unless some further inadmissible evidence E overwrites this. For example, if the agent has evidence that a coin is biased such that the chance of H is 0.8, then he should choose the credence of H to be 0.8, unless he has some further evidence that overwrites this choice, for example, when the toss already happened and he knows the result. Without further such evidence, for example, just before the coin is tossed, the reasonable credence to be assigned to H is 0.8.

Suppose now the agent learned that ${\mathrm{ch}}_t\left(A\right)=r$ and updated his credence function accordingly to $C^{\prime }$:

(5)$C^{\prime }\left(B\right|E)≔C(B\left|{\mathrm{ch}}_t\right(A)=r\wedge E)$

for all B. That is, the agent added ${\mathrm{ch}}_t\left(A\right)=r$ to the set of evidences. Is it true that this new credence function automatically satisfies the Principal Principle for any ${\mathrm{ch}}_t\left(B\right)=r^{\prime }$ condition, or are additional assumptions needed?

Just from assuming that C satisfies the Principal Principle without admissible E events, it does not follow that $C^{\prime }$ also satisfies the Principal Principle. However, it does follow from the Principal Principle with admissible E events included, as long as events of the form ${\mathrm{ch}}_t\left(A\right)=r$ are admissible. And indeed, Lewis insists that such events must be admissible. Then, applying the Principal Principle to ${\mathrm{ch}}_t\left(A\right)=r\wedge E$ instead of E, we have

(6)$C^{\prime }\left(B\left|{\mathrm{ch}}_t\left(B\right)=r^{\prime }\wedge E)=C(B\right|{\mathrm{ch}}_t\right(B)=r^{\prime }\wedge ({\mathrm{ch}}_t\left(A\right)=r\wedge E\left)\right)=r^{\prime }\mathrm{.}$

Now consider this: remember, equation (2) says, for any admissible evidence E, $C\left(A\right|E)={\sum }_{r}r\dot{}C({\mathrm{ch}}_t\left(A\right)=r\left|E\right)$, where the sum runs through all r values that ${\mathrm{ch}}_t\left(A\right)$ can possibly take. We have seen that this follows from the Principal Principle if the credence function obeys the laws of probability. The other direction is also true: the assumption that the credence C satisfies the laws of probability and that events of the form ${\mathrm{ch}}_t\left(A\right)=r$ are admissible, together with (2), implies the Principal Principle. To see this, consider that

$\begin{array}{}C\left(A\right|{\mathrm{ch}}_t\left(A\right)=r\wedge E)=\sum _{r^{\prime }}{r}^{\prime }\dot{}C({\mathrm{ch}}_t\left(A\right)=r^{\prime }\left|{\mathrm{ch}}_t\right(A)=r\wedge E),\end{array}$

where we applied equation (2) not for E but for ${\mathrm{ch}}_t\left(A\right)=r\wedge E$. But

$\sum _{r^{\prime }}{r}^{\prime }\dot{}C\left({\mathrm{ch}}_t\right(A)=r^{\prime }|{\mathrm{ch}}_t\left(A\right)=r\wedge E)=r$

for any r, because if C satisfies the laws of probability, then $C\left({\mathrm{ch}}_t\right(A)=r^{\prime }|{\mathrm{ch}}_t\left(A\right)=r\wedge E)=1$ if $r^{\prime }=r$ and $C\left({\mathrm{ch}}_t\right(A)=r^{\prime }|{\mathrm{ch}}_t\left(A\right)=r\wedge E)=0$ if $r^{\prime }\ne r$ no matter what E is. With this, equation (5) becomes

$\begin{array}{}C\left(A\right|{\mathrm{ch}}_t\left(A\right)=r\wedge E)=r\mathrm{.}\end{array}$

Hence, given that the laws of probability are satisfied and that events of the form ${\mathrm{ch}}_t\left(A\right)=r$ are admissible, the Principal Principle and the validity of equation (2) for all admissible E are in fact equivalent. This is an important observation, because as (2) implies the Principal Principle, it gives us a hint how to construct in the sections below the extension Rédei and Gyenis require for their consistency notion: namely, if the credence of each chancy event A is computed from the credences assigned to the possible values of the objective chance of A using equation (2), then the credence function on the chances and on the chancy events satisfies the Principal Principle.

3. Consistency Questions of Rédei and Gyenis

Rédei and Gyenis (Reference Rédei and Gyenis2013) raised the question what it really meant to conditionalize with the event that the chance of something is a certain value. They posed their question in a more abstract and mathematically well-defined way than how we have proceeded so far. It is essentially as follows.

“Chance” in their terminology is replaced by a probability measure $p_{\mathrm{obj}}$ over an algebra $S_{\mathrm{obj}}$ representing the chancy events (corresponding to H and T in the example). They drop time t to make the discussion simpler, as it is not necessary for presenting their complaints. Credence is replaced by a probability measure $p_{\mathrm{subj}}$ over some algebra $S_{\mathrm{subj}}$. They observe that it is necessary that $S_{\mathrm{subj}}$ includes $S_{\mathrm{obj}}$, as we want to associate credence at least to the real events. Furthermore, $S_{\mathrm{subj}}$ has to contain events of the form $⌜P_{\mathrm{obj}}\left(A\right)=r⌝$, meaning the event that “the objective chance of A equals r.” I intentionally do not follow their notation $⌜p_{\mathrm{obj}}\left(A\right)=r⌝$ because I want to be able to write $⌜P_{\mathrm{obj}}\left(A\right)=p_{\mathrm{obj}}\left(A\right)⌝$ meaning “the objective chance of A equals $p_{\mathrm{obj}}\left(A\right)$,” where $p_{\mathrm{obj}}\left(A\right)$ is a probability measure on $S_{\mathrm{obj}}$. Rédei and Gyenis define the abstract Principal Principle:

Definition 1. The subjective probabilities $p_{\mathrm{subj}}\left(A\right)$ are related to the objective probabilities $p_{\mathrm{obj}}\left(A\right)$, as

(7)$\begin{array}{}{p}_{\mathrm{subj}}\left(A\right|⌜P_{\mathrm{obj}}\left(A\right)=p_{\mathrm{obj}}\left(A\right)⌝)=p_{\mathrm{obj}}(A),\end{array}$

as long as the conditioning makes sense.Footnote ¹

Rédei and Gyenis have another notion too, the stable abstract Principal Principle, which also requires that given any $A\in S_{\mathrm{obj}}$, for all $B\in S_{\mathrm{obj}}$,

$p_{\mathrm{subj}}\left(A\right|⌜P_{\mathrm{obj}}\left(A\right)=p_{\mathrm{obj}}\left(A\right)⌝)=p_{\mathrm{subj}}(A|⌜P_{\mathrm{obj}}(A)=p_{\mathrm{obj}}(A)⌝\cap ⌜P_{\mathrm{obj}}(B)=p_{\mathrm{obj}}(B)⌝),$

as long as the conditional probabilities make sense. In other words, learning the objective probability of B should not destroy the Principal Principle for A.

Rédei and Gyenis also define their strong consistency condition:

Clearly, $A^{\prime }$ would be the event “the objective probability of A is $p_{\mathrm{obj}}\left(A\right)$.”

Rédei and Gyenis, then, have two claims:

• • In order for the (abstract) Principal Principle to be meaningful, it is necessary that strong consistency hold, and strong consistency is tacitly assumed.

• • It is not clear whether strong consistency holds.

I strongly disagree with the first claim. Lewis’s (Reference Lewis1980) work is titled “A Subjectivist’s Guide to Objective Chance.” That is, chance is viewed as being objective, and credence can be associated to statements about objective chance. When the objective world contains both the events and their chances, credence is not an arbitrary probability measure, but the Principal Principle limits what a reasonable credence function may look like. Coming back to Lewis’s example we cited at the beginning, given the credences assigned to the various biases, it does not make sense to assign 50% credence to heads, but it has to be 62%. In other words, the Principal Principle is a consistency condition itself for the case when we assign credences to statements about objective chance. That is, there is only one $S_{\mathrm{obj}}$; for a subjectivist who believes in objective chance, statements about objective chance are in $S_{\mathrm{obj}}$ to start with, and the Principal Principle is the consistency condition that we have to require for a credence function on $S_{\mathrm{obj}}$.

This view that I proposed in the previous paragraph I believe is also supported by the usual Bayesian analysis. The Bayesian prior, which corresponds to credence, is given not on the real outcomes such as heads and tails but instead on the set of probability measures (described by some parameter) over the real outcomes. In this set of probability measures, of course, events expressing that the probability of a chancy event A is r can be represented by the set of the form $\{p:p(A)=r\}$. That is, in Bayesian analysis as well, statements about the values of these probabilities are part of the event space from the beginning. Prior probabilities of the chancy events are computed from the prior measure over the probabilities similar to formula (1), just as Lewis did his computation.

However, if we insist that $S_{\mathrm{obj}}$ denotes only the chancy events and that it should be extended, then from the above, it is clear how to do it: we take all probability measures on $S_{\mathrm{obj}}$ and define an event algebra on it that is fine enough to include all sets of the form $\{p:p(A)=r\}$, where $A\in S_{\mathrm{obj}}$ and $r\in [0,1]$. Let us call this event algebra over the probabilities $S_{\mathrm{obj}}^{\mathrm{ch}}$ (this is the algebra of chances). We can define $S_{\mathrm{subj}}≔S_{\mathrm{obj}}\otimes S_{\mathrm{obj}}^{\mathrm{ch}}$ (i.e., the algebra generated by elements of the form $A_1\times A_2$, where $A_1\in S_{\mathrm{obj}}$ and $A_2\in S_{\mathrm{obj}}^{\mathrm{ch}}$). But even in such a setup, an agent would not start from a credence function on $S_{\mathrm{obj}}$ that he would want to extend to $S_{\mathrm{subj}}$. Instead, he would start fixing a credence function on the chances—or, with Bayesian language, he would first fix a Bayesian prior $p_{\mathrm{subj}}^{\mathrm{ch}}$ on $S_{\mathrm{obj}}^{\mathrm{ch}}$—and would obtain the prior probability $p_{\mathrm{subj}}^0$ on $S_{\mathrm{obj}}$ using the usual formula corresponding to (1):

$p_{\mathrm{subj}}^0\left(A\right)≔\sum _{r}r\dot{}{p}_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=r\left\}\right),$

which is just the same as the expected value of the function $p\mapsto p\left(A\right)$ with respect to the probability $p_{\mathrm{subj}}^{\mathrm{ch}}$.Footnote ² Furthermore, the joint probabilities on $S_{\mathrm{subj}}$ would also be the usual:Footnote ³

$p_{\mathrm{subj}}(A_1\times A_2)≔\sum _{r}r\cdot p_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A_1\right)=r\wedge p\in A_2\left\}\right)\mathrm{.}$

That is exactly what happens in Lewis’s example about the coin tosses: in that case $S_{\mathrm{obj}}$ can be taken to have two atomic elements, heads and tails, and $S_{\mathrm{obj}}^{\mathrm{ch}}$ to have three atomic elements with nonzero credence (Bayesian prior):

• • one putting 50% chance on heads with credence 27%.

• • one putting 35% chance on heads with credence 22%.

• • one putting 80% chance on heads with credence 51%.

Then the credence can be extended to $S_{\mathrm{obj}}$ giving 62% for heads and 38% for tails. It is not the 62% and the 38% that an agent postulates first and then finds a measure on $S_{\mathrm{obj}}^{\mathrm{ch}}$ (which would correspond to the suggestion of Rédei and Gyenis), but the other way.

About the second point Rédei and Gyenis made, I do agree. It is not obvious, but still it turns out to be essentially true, which is the topic of the rest of this discussion.

4. Strong Consistency of the (Abstract) Principal Principle

Even if strong consistency does not seem to us necessary to be required in general, in certain cases it may be interesting to fix credences on the set of objective events and then try to extend it to a larger algebra that includes also events like $⌜P_{\mathrm{obj}}\left(A\right)=r⌝$. Such an extension was, for example, considered by Diaconis and Zabell (Reference Diaconis and Zabell1982, sec. 2.1) without the strong consistency condition.

Here we prove the following strong consistency theorem:

Proof. Consider the set of all probability measures on $(X_{\mathrm{obj}},S_{\mathrm{obj}})$: denote it by $X_{\mathrm{obj}}^{\mathrm{ch}}$ (this is the set of all possible chances):

$X_{\mathrm{obj}}^{\mathrm{ch}}≔\{p:p\text{is}\text{a}\text{probability}\text{measure}\text{on}{(X}_{\text{obj}}\text{,}{S}_{\text{obj}})\}\mathrm{.}$

For an $A\in S_{\mathrm{obj}}$ and an $r\in [0,1]$, let us define

$⌜P_{\mathrm{obj}}\left(A\right)=r{⌝}^{\mathrm{ch}}≔\{p:p(A)=r\}\subseteq X_{\mathrm{obj}}^{\mathrm{ch}}\mathrm{.}$

Take a fine enough σ-algebra $S_{\mathrm{obj}}^{\mathrm{ch}}$ on $X_{\mathrm{obj}}^{\mathrm{ch}}$ such that for all $A\in S_{\mathrm{obj}}$ and $r\in [0,1]$, $⌜P_{\mathrm{obj}}\left(A\right)=r{⌝}^{\mathrm{ch}}\in S_{\mathrm{obj}}^{\mathrm{ch}}$.

I claim that it is possible to find a measure $p_{\mathrm{subj}}^{\mathrm{ch}}$ (credence of chances) on $S_{\mathrm{obj}}^{\mathrm{ch}}$ such that $p_{\mathrm{subj}}^{\mathrm{ch}}({\cap }_{i=1}^n⌜P_{\mathrm{obj}}(A_i)=p_{\mathrm{obj}}(A_i\left){⌝}^{\mathrm{ch}}\right)\ne 0$ for any $A_1,\dots ,A_n\in S_{\mathrm{obj}}$ (in order to be able to take conditionals), and for all $A\in S_{\mathrm{obj}}$,

$p_{\mathrm{subj}}^0\left(A\right)=\sum _{r}r\dot{}{p}_{\mathrm{subj}}^{\mathrm{ch}}(⌜P_{\mathrm{obj}}(A)=r{⌝}^{\mathrm{ch}}),$

where the summation is taken over all values r for which $p_{\mathrm{subj}}^{\mathrm{ch}}(⌜P_{\mathrm{obj}}(A)=r{⌝}^{\mathrm{ch}})\ne 0$ (it could also be defined as an integral over $\left[0\mathrm{,}1\right]$; see n. 2). For a $p\in X_{\mathrm{obj}}^{\mathrm{ch}}$, let ${\delta }_p$ denote a Dirac measure concentrated on p. That is, for any $S\in S_{\mathrm{obj}}^{\mathrm{ch}}$, ${\delta }_p\left(S\right)=1$ if and only if $p\in S$, and ${\delta }_p\left(S\right)=0$ if and only if $p\overline{)\in }S$. We prove that $p_{\mathrm{subj}}^{\mathrm{ch}}$ can simply be taken to be of the form

$p_{\mathrm{subj}}^{\mathrm{ch}}=a{\dot{}\delta }_{p_{\mathrm{obj}}}+b\dot{}{\delta }_{p_0},$

where a and b are conveniently chosen constants and $p_0$ is a probability measure on $(X_{\mathrm{obj}},S_{\mathrm{obj}})$: we assumed that there is an $a\in (0,1)$ such that for all $A\in S_{\mathrm{obj}}$, $a\dot{}{p}_{\mathrm{obj}}\left(A\right)<p_{\mathrm{subj}}^0\left(A\right)$ or $p_{\mathrm{obj}}\left(A\right)=p_{\mathrm{subj}}^0\left(A\right)=0$. Clearly, the measure $a\dot{}{\delta }_{p_{\mathrm{obj}}}$ is a positive measure on $(X_{\mathrm{obj}}^{\mathrm{ch}},S_{\mathrm{obj}}^{\mathrm{ch}})$, but as $a\in (0,1)$, it is smaller than a probability measure. Let $b≔1-a$, and let $p_0≔(1/1-a)\dot{}(p_{\mathrm{subj}}^0-a\dot{}{p}_{\mathrm{obj}})$. As $a\in (0,1)$ and $a\dot{}{p}_{\mathrm{obj}}<p_{\mathrm{subj}}^0$, $p_0$ is a positive measure on $(X_{\mathrm{obj}},S_{\mathrm{obj}})$. As $p_0\left(X_{\mathrm{obj}}\right)=(1/1-a)\dot{}(1-a)=1$, this is a probability measure on $(X_{\mathrm{obj}},S_{\mathrm{obj}})$ and hence an element of $X_{\mathrm{obj}}^{\mathrm{ch}}$. Let

$p_{\mathrm{subj}}^{\mathrm{ch}}≔a\dot{}{\delta }_{p_{\mathrm{obj}}}+b\dot{}{\delta }_{p_0}\mathrm{.}$

Since $a+b=1$, this is also a probability measure. For any $A_1,\dots ,A_n\in S_{\mathrm{obj}}$,

$\begin{array}{cc}{p}_{\mathrm{subj}}^{\mathrm{ch}}(\underset{i=1}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}(A_i)=p_{\mathrm{obj}}(A_i\left){⌝}^{\mathrm{ch}}\right)& =p_{\mathrm{subj}}^{\mathrm{ch}}\left(\underset{i=1}{\overset{n}{\cap }}\right\{p:p\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right)\left\}\right)\\ & =a\dot{}{\delta }_{p_{\mathrm{obj}}}\left(\underset{i=1}{\overset{n}{\cap }}\right\{p:p\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right)\left\}\right)+b\dot{}{\delta }_{p_0}\left(\underset{i=1}{\overset{n}{\cap }}\right\{p:p\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right)\left\}\right)\\ & =a+b\dot{}{\delta }_{p_0}\left(\underset{i=1}{\overset{n}{\cap }}\right\{p:p\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right)\left\}\right)\\ & \ge a>0,\end{array}$

where we used the fact that $p_{\mathrm{obj}}\in {\cap }_{i=1}^n\{p:p(A_i)=p_{\mathrm{obj}}(A_i\left)\right\}$ and that ${\delta }_{p_{\mathrm{obj}}}\left(S\right)=1$ if and only if $p_{\mathrm{obj}}\in S$. We also have that for all $A\in S_{\mathrm{obj}}$,

$\begin{array}{c}\sum _{r}r\dot{}{p}_{\mathrm{subj}}^{\mathrm{ch}}(⌜P_{\mathrm{obj}}(A)=r{⌝}^{\mathrm{ch}})\\ =\sum _{r}r\dot{}(a\dot{}{\delta }_{p_{\mathrm{obj}}}(\{p:p(A)=r\})+b\dot{}{\delta }_{p_0}(\{p:p(A)=r\}\left)\right)\\ =p_{\mathrm{obj}}\left(A\right)\dot{}a+p_0\left(A\right)\dot{}b\\ =p_{\mathrm{obj}}\left(A\right)\dot{}a+\frac{1}{1-a}\cdot \left(p_{\mathrm{subj}}^0\right(A)-a\dot{}{p}_{\mathrm{obj}}(A\left)\right)\dot{}(1-a)\\ =p_{\mathrm{obj}}\left(A\right)\dot{}a+p_{\mathrm{subj}}^0\left(A\right)-a\dot{}{p}_{\mathrm{obj}}\left(A\right)=p_{\mathrm{subj}}^0\left(A\right)\mathrm{.}\end{array}$

Here, note that $p_{\mathrm{subj}}^{\mathrm{ch}}(⌜P_{\mathrm{obj}}(A)=r{⌝}^{\mathrm{ch}})\ne 0$ for at most two different r’s—$r=p_{\mathrm{obj}}\left(A\right)$ and $r=p_0\left(A\right)$—so there is no problem using the summation sign.

Let $(X_{\mathrm{subj}},S_{\mathrm{subj}})≔(X_{\mathrm{obj}}\times X_{\mathrm{obj}}^{\mathrm{ch}},S_{\mathrm{obj}}\otimes S_{\mathrm{obj}}^{\mathrm{ch}}$). Here, $S_{\mathrm{obj}}\otimes S_{\mathrm{obj}}^{\mathrm{ch}}$ is the σ-algebra generated by elements of the form $A\times E$, where $A\in S_{\mathrm{obj}}$ and $E\in S_{\mathrm{obj}}^{\mathrm{ch}}$. Let the measure $p_{\mathrm{subj}}$ on $(X_{\mathrm{subj}},S_{\mathrm{subj}})$ be generated by

$p_{\mathrm{subj}}(A\times E)≔\sum _{r}r\dot{}{p}_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=r\wedge p\in E\left\}\right)$

for all $A\in S_{\mathrm{obj}}$ and $E\in S_{\mathrm{obj}}^{\mathrm{ch}}$. Again, because of the way we defined $p_{\mathrm{subj}}^{\mathrm{ch}}$, for any $A\times E$, there are at most two values of r ($p_{\mathrm{obj}}\left(A\right)$ and $p_0\left(A\right)$) such that the summands in the above summation are nonzero, so it is not incorrect to keep using the summation instead of an integral.

For all $A\in S_{\mathrm{obj}}$, let

$h\left(A\right)≔A\times X_{\mathrm{obj}}^{\mathrm{ch}}$

and

$⌜P_{\mathrm{obj}}\left(A\right)=r⌝≔X_{\mathrm{obj}}\times \{p:p(A)=r\}=X_{\mathrm{obj}}\times ⌜P_{\mathrm{obj}}\left(A\right)=r{⌝}^{\mathrm{ch}}$

, and $A^{\prime }≔⌜P_{\mathrm{obj}}\left(A\right)=p_{\mathrm{obj}}\left(A\right)⌝$.

Clearly,

$\begin{array}{cc}{p}_{\mathrm{subj}}\left(h\right(A\left)\right)& =p_{\mathrm{subj}}(A\times X_{\mathrm{obj}}^{\mathrm{ch}})\\ & =\sum _{r}r\dot{}{p}_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=r\wedge p\in X_{\mathrm{obj}}^{\mathrm{ch}}\left\}\right)\\ & =\sum _{r}r\dot{}{p}_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=r\left\}\right)\\ & =\sum _{r}r\dot{}{p}_{\mathrm{subj}}^{\mathrm{ch}}(⌜P_{\mathrm{obj}}(A)=r{⌝}^{\mathrm{ch}})\\ & =p_{\mathrm{subj}}^0\left(A\right)\end{array}$

by the choice of $p_{\mathrm{subj}}^{\mathrm{ch}}$.

Then, for $A_1,\dots ,A_n\in S_{\mathrm{obj}}$, we have

$\begin{array}{cc}{p}_{\mathrm{subj}}\left(\underset{i=1}{\overset{n}{\cap }}{A}_i^{\prime }\right)& =p_{\mathrm{subj}}(\underset{i=1}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}(A_i)=p_{\mathrm{obj}}(A_i)⌝)\\ & =p_{\mathrm{subj}}\left(\underset{i=1}{\overset{n}{\cap }}\right(X_{\mathrm{obj}}\times ⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left)\right)\\ & =p_{\mathrm{subj}}^{\mathrm{ch}}(\underset{i=1}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}(A_i)=p_{\mathrm{obj}}(A_i\left){⌝}^{\mathrm{ch}}\right)>0,\end{array}$

as we have seen.

To see iv, fix $A,A_1,\dots ,A_n\in S_{\mathrm{obj}}$, and for simplicity let $A_0≔A$. By applying the above definitions, and as $p_{\mathrm{subj}}\left({\cap }_{i=0}^n{A}_i^{\prime }\right)>0$,

$\begin{array}{c}{p}_{\mathrm{subj}}\left(h\right(A\left)\right|A^{\prime }\cap \underset{i=1}{\overset{n}{\cap }}{A}_i^{\prime })\\ =\frac{p_{\mathrm{subj}}\left(h\right(A)\cap \underset{i=0}{\overset{n}{\cap }}{A}_i^{\prime })}{p_{\mathrm{subj}}\left(\underset{i=0}{\overset{n}{\cap }}{A}_i^{\prime }\right)}\\ =\frac{p_{\mathrm{subj}}\left(\right(A\times X_{\mathrm{obj}}^{\mathrm{ch}})\cap \underset{i=0}{\overset{n}{\cap }}(X_{\mathrm{obj}}\times ⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left)\right)}{p_{\mathrm{subj}}\left(\underset{i=0}{\overset{n}{\cap }}\right(X_{\mathrm{obj}}\times ⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left)\right)}\\ =\frac{p_{\mathrm{subj}}\left(\right(A\times X_{\mathrm{obj}}^{\mathrm{ch}})\cap (X_{\mathrm{obj}}\times \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left)\right)}{p_{\mathrm{subj}}(X_{\mathrm{obj}}\times \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}(A_i)=p_{\mathrm{obj}}(A_i\left){⌝}^{\mathrm{ch}}\right)}\\ =\frac{p_{\mathrm{subj}}(A\times \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}(A_i)=p_{\mathrm{obj}}(A_i\left){⌝}^{\mathrm{ch}}\right)}{p_{\mathrm{subj}}(X_{\mathrm{obj}}\times \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}(A_i)=p_{\mathrm{obj}}(A_i\left){⌝}^{\mathrm{ch}}\right)}\\ =\frac{\sum _{r}r\cdot p_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=r\wedge p\in \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left\}\right)}{\sum _{r}r\cdot p_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(X_{\mathrm{obj}}\right)=r\wedge p\in \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left)\right\})}\\ =\sum _{r}r\dot{}\frac{p_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=r\wedge p\in \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left\}\right)}{p_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(X_{\mathrm{obj}}\right)=1\wedge p\in \underset{i=0}{\overset{n}{\cap }}⌜P_{\mathrm{obj}}\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right){⌝}^{\mathrm{ch}}\left)\right\})}\\ =\sum _{r}r\frac{p_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=r\wedge p\left(A\right)=p_{\mathrm{obj}}\left(A\right)\wedge \underset{i=1}{\overset{n}{\wedge }}p\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right)\left\}\right)}{p_{\mathrm{subj}}^{\mathrm{ch}}\left(\right\{p:p\left(A\right)=p_{\mathrm{obj}}\left(A\right)\wedge \underset{i=1}{\overset{n}{\wedge }}p\left(A_i\right)=p_{\mathrm{obj}}\left(A_i\right)\left\}\right)}\\ =p_{\mathrm{obj}}\left(A\right)\end{array}$

because the numerator is only nonzero when $r=p_{\mathrm{obj}}\left(A\right)$, in which case the numerator and the denominator are equal. This gives us iv. For the special case when there is only one $A_1$, namely, $X_{\mathrm{obj}}$, we obtain the proof of i.

Finally, consider ii. In the construction so far, for $A\ne B$, $A^{\prime }\ne B^{\prime }$ is actually not ensured. But it is easy to modify $X_{\mathrm{subj}}$ so that this condition is ensured as well. Consider the disjoint union $X_{\mathrm{subj}}\uplus S_{\mathrm{obj}}$, set $p_{\mathrm{subj}}\left(S_{\mathrm{obj}}\right)≔0$, and define $A^{\prime }≔⌜P_{\mathrm{obj}}\left(A\right)=p_{\mathrm{obj}}\left(A\right)⌝\cup \left\{A\right\}$. This will not interfere with the probabilities, but including $\left\{A\right\}$ ensures that $A\ne B$ implies $A^{\prime }\ne B^{\prime }$. QED

If $X_{\mathrm{obj}}$ (and hence $S_{\mathrm{obj}}$) is finite, and if $p_{\mathrm{obj}}$ is absolutely continuous with respect to $p_{\mathrm{subj}}^0$ (i.e., for any $A\in S_{\mathrm{obj}}$, $p_{\mathrm{obj}}\left(A\right)\ne 0$ implies $p_{\mathrm{subj}}^0\left(A\right)\ne 0$), then for any

$a\in (0,{\min }_{A\in S_{\mathrm{obj}}\wedge p_{\mathrm{obj}}\left(A\right)>0}\frac{p_{\mathrm{subj}}^0\left(A\right)}{p_{\mathrm{obj}}\left(A\right)})\subseteq \left(0\mathrm{,}1\right)$

we have that for all $A\in S_{\mathrm{obj}}$, $a\dot{}{p}_{\mathrm{obj}}\left(A\right)<p_{\mathrm{subj}}^0\left(A\right)$ or $p_{\mathrm{obj}}\left(A\right)=p_{\mathrm{subj}}^0\left(A\right)=0$, and the conclusions of the theorem hold. Hence, we have