Proof Observe that T is at most single-valued. On the other hand, Lemma 2.2(v) states that there exists a sequence $(u_n)_{n\in \mathbb {N}}$ in $\mathfrak {H}$ such that

(3.5)

(i)(a) $\Rightarrow $ (i)(b): Let be such that $\mu (\Xi )=0$ and, for every $\omega \in \complement\!\Xi $ , $\mathsf {T}_{\omega }$ is $\beta $ -Lipschitzian. Then

(3.6)

In turn, since $\mathfrak {G}$ is a vector subspace of $\prod _{\omega \in \Omega }{\mathsf {H}}_{\omega }$ , we infer from [A] and (1.6) that, for every $x\in \mathfrak {H}$ and every $y\in \mathfrak {H}$ , the mapping $\omega \mapsto \mathsf {T}_{\omega }(x(\omega ))-\mathsf {T}_{\omega }(y(\omega ))$ lies in $\mathfrak {H}$ . Thus, [B] implies that, for every $x\in \mathfrak {H}$ , the mapping $\omega \mapsto \mathsf {T}_{\omega }(x(\omega ))$ lies in $\mathfrak {H}$ as the sum of two mappings in $\mathfrak {H}$ , namely $\omega \mapsto \mathsf {T}_{\omega }(x(\omega ))-\mathsf {T}_{\omega }(z(\omega ))$ and $\omega \mapsto \mathsf {T}_{\omega }(z(\omega ))$ . Therefore, $\operatorname { {dom}} T=\mathcal {H}$ . Additionally, it results from (3.6) and (1.7) that T is $\beta $ -Lipschitzian.

(i)(b) $\Rightarrow $ (i)(a): Fix temporarily $n\in \mathbb {N}$ and $m\in \mathbb {N}$ . For every such that $\mu (\Xi )<{+}\infty $ , since $1_{\Xi }u_n\in \mathfrak {H}$ and $1_{\Xi }u_m\in \mathfrak {H}$ thanks to Lemma 2.2(iii), we derive from (1.7) that

(3.7)

Hence, since is $\sigma $ -finite, there exists such that

(3.8)

Now set $\Xi =\bigcup _{n\in \mathbb {N},m\in \mathbb {N}}\Xi _{n,m}$ , let $\omega \in \complement\!\Xi $ , let $\mathsf {x}\in {\mathsf {H}}_{\omega }$ , and let $\mathsf {y}\in {\mathsf {H}}_{\omega }$ . Then, with $\mu (\Xi )=0$ and, in view of (3.5), there exist sequences $(k_n)_{n\in \mathbb {N}}$ and $(l_n)_{n\in \mathbb {N}}$ in $\mathbb {N}$ such that $u_{k_n}(\omega )\to \mathsf {x}$ and $u_{l_n}(\omega )\to \mathsf {y}$ . At the same time, by (3.8),

(3.9)

Thus, since is weakly lower semicontinuous, letting $n\to {+}\infty $ and invoking the strong-to-weak continuity of $\mathsf {T}_{\omega }$ , we get .

(ii) and (iii): Argue as in (i).

Proof Set . We derive from Definition 1.3 and [Reference Bauschke and Combettes3, Proposition 23.2(ii)] that

(3.12) $$ \begin{align} (\forall x\in\mathcal{H})\quad Tx &=\bigl\{{p\in\mathcal{H}}\mid{(\forall^{\mu}\omega\in\Omega)\,\, p(\omega)\in J_{\gamma{\mathsf{A}}_{\omega}}\big({x(\omega)}\big)}\bigr\} \nonumber\\ &=\bigl\{{p\in\mathcal{H}}\mid{(\forall^{\mu}\omega\in\Omega)\,\, \gamma^{-1}\big({x(\omega)-p(\omega)}\big)\in{\mathsf{A}}_{\omega}\big({p(\omega)}\big)}\bigr\} \nonumber\\ &=\bigl\{{p\in\mathcal{H}}\mid{\gamma^{-1}(x-p)\in Ap}\bigr\} \nonumber\\ &=J_{\gamma A}x. \end{align} $$

Likewise, upon setting , we deduce from Definition 1.3 and [Reference Bauschke and Combettes3, Proposition 23.2(iii)] that

(3.13)

which completes the proof.

Proof We infer from Assumption 3.6[A] and [Reference Bauschke and Combettes3, Propositions 20.22 and 23.20] that, for every $\omega \in \Omega $ , ${\mathsf {A}}_{\omega }^{-1}$ is maximally monotone and $J_{{\mathsf {A}}_{\omega }^{-1}}={\mathrm {Id}}_{{\mathsf {H}}_{\omega }}-J_{{\mathsf {A}}_{\omega }}$ . In turn, for every $x\in \mathfrak {H}$ , since $\mathfrak {G}$ is a vector subspace of $\prod _{\omega \in \Omega }{\mathsf {H}}_{\omega }$ , it follows from Assumption 3.6[B] that the mapping $\omega \mapsto J_{{\mathsf {A}}_{\omega }^{-1}}(x(\omega ))$ lies in $\mathfrak {G}$ as the difference of the mappings x and $\omega \mapsto J_{{\mathsf {A}}_{\omega }}(x(\omega ))$ . Finally, Proposition 3.2(iv) and Assumption 3.6[C] yield .

Proof (i): By [Reference Bauschke and Combettes3, Proposition 23.2(i)] and Assumption 3.6[C], $\operatorname { {ran}} J_A=\operatorname { {dom}} A\neq \varnothing $ and there thus exist z and r in $\mathcal {H}$ such that $r\in J_Az$ or, equivalently, $z-r\in Ar$ . Hence, for $\mu $ -almost every $\omega \in \Omega $ , $z(\omega )-r(\omega )\in {\mathsf {A}}_{\omega }(r(\omega ))$ and, therefore, the monotonicity of ${\mathsf {A}}_{\omega }$ yields $r(\omega )=J_{{\mathsf {A}}_{\omega }}(z(\omega ))$ . Thus, because $r\in \mathfrak {H}$ , we infer from Lemma 2.2(ii) that the mapping $\omega \mapsto J_{{\mathsf {A}}_{\omega }}(z(\omega ))$ lies in $\mathfrak {H}$ . In turn, appealing to Assumption 3.6[B], we deduce from Proposition 3.4(iii) (applied to the firmly nonexpansive operators $(J_{{\mathsf {A}}_{\omega }})_{\omega \in \Omega }$ ) and Proposition 3.5 that $J_A\colon \mathcal {H}\to \mathcal {H}$ is firmly nonexpansive. Consequently, [Reference Bauschke and Combettes3, Proposition 23.8(iii)] guarantees that A is maximally monotone.

(ii): Use (i), Proposition 3.5, and Lemma 2.2(ii).

(iii): By (i) and [Reference Bauschke and Combettes3, Corollary 21.14], $\operatorname { {\overline {dom}}} A$ is a nonempty closed convex subset of $\mathcal {H}$ . Fix temporarily $x\in \mathfrak {H}$ , let $(\gamma _n)_{n\in \mathbb {N}}$ be a sequence in $]{0},{1}[$ such that $\gamma _n\downarrow 0$ , and set

(3.15) $$ \begin{align} p=\operatorname{{proj}}_{\operatorname{{\overline{dom}}} A}x \quad\text{and}\quad (\forall n\in\mathbb{N})\;\; p_n\colon\omega\mapsto J_{\gamma_n{\mathsf{A}}_{\omega}}\big({x(\omega)}\big). \end{align} $$

We infer from (ii)(a) that, for every $n\in \mathbb {N}$ , $p_n\in \mathfrak {H}$ and $p_n=J_{\gamma _nA}x$ . Thus, it follows from (i) and [Reference Bauschke and Combettes3, Theorem 23.48] that $p_n\to p$ in $\mathcal {H}$ . In turn, Lemma 2.3 ensures that there exist a strictly increasing sequence $(k_n)_{n\in \mathbb {N}}$ in $\mathbb {N}$ and a set such that $\mu (\Xi )=0$ and $(\forall \omega \in \complement\!\Xi ) p_{k_n}(\omega )\to p(\omega )$ . On the other hand, we deduce from Assumption 3.6[A] and [Reference Bauschke and Combettes3, Theorem 23.48] that $(\forall \omega \in \complement\!\Xi ) p_{k_n}(\omega )=J_{\gamma _{k_n}{\mathsf {A}}_{\omega }}(x(\omega )) \to \operatorname { {proj}}_{\operatorname { {\overline {dom}}}{\mathsf {A}}_{\omega }}(x(\omega ))$ . Therefore, $(\forall \omega \in \complement\!\Xi ) p(\omega )=\operatorname { {proj}}_{\operatorname { {\overline {dom}}}{\mathsf {A}}_{\omega }}(x(\omega ))$ . Hence, because $p\in \mathfrak {H}$ , it results from Lemma 2.2(ii) that the mapping $\omega \mapsto \operatorname { {proj}}_{\operatorname { {\overline {dom}}}{\mathsf {A}}_{\omega }}(x(\omega ))$ is a representative in $\mathfrak {H}$ of $\operatorname { {proj}}_{\operatorname { {\overline {dom}}} A}x$ . This confirms that

(3.16)

Therefore, using Definition 3.1, we get

(3.17)

Thus, is a closed subset of $\mathcal {H}$ . Consequently, we deduce from Proposition 3.2(i) and Definition 3.1 that

(3.18)

which furnishes the desired identities.

(iv): In the light of Proposition 3.2(iv) and Proposition 3.7, the claim follows from (iii) applied to the family $({\mathsf {A}}_{\omega }^{-1})_{\omega \in \Omega }$ .

(v): Let $(\gamma _n)_{n\in \mathbb {N}}$ be a sequence in $]{0},{1}[$ such that $\gamma _n\downarrow 0$ , and set

(3.19)

Then, on account of (ii)(b),

(3.20)

(v)(a): For every $\omega \in \Omega $ , since ${\mathsf {A}}_{\omega }$ is maximally monotone and $x(\omega )\in \operatorname { {dom}}{\mathsf {A}}_{\omega }$ , [Reference Bauschke and Combettes3, Corollary 23.46(i)] yields $p_n(\omega )\to p(\omega )$ . Hence, thanks to Lemma 2.2(iv), we obtain $p\in \mathfrak {G}$ .

(v)(b): Set . It follows from (3.20), (i), and [Reference Bauschke and Combettes3, Corollary 23.46(i)] that $p_n\to q$ in $\mathcal {H}$ . Thus, we infer from Lemma 2.3 that there exists a strictly increasing sequence $(k_n)_{n\in \mathbb {N}}$ in $\mathbb {N}$ such that $(\forall ^{\mu }\omega \in \Omega ) p_{k_n}(\omega )\to q(\omega )$ . In turn, $p=q\ \mu \text {-a.e.}$ and we conclude by invoking Lemma 2.2(ii).

Proof In the light of Example 2.1(iv), $\mathcal {H}$ is the Hilbert direct integral of the -measurable vector field $(({\mathsf {H}}_{\omega })_{\omega \in \Omega },\mathfrak {G})$ defined by

(3.23)

Additionally, by (3.22),

(3.24)

(i) $\Rightarrow $ (ii): We have $\operatorname { {dom}} A\neq \varnothing $ and $J_A\colon \mathcal {H}\to \mathcal {H}$ [Reference Bauschke and Combettes3, Corollary 23.11(i)]. Thus, invoking Proposition 3.5 and Lemma 2.2(ii), we deduce that

(3.25)

Next, take $\mathsf {x}\in \mathsf {H}$ . Since is $\sigma $ -finite, there exists an increasing sequence $(\Omega _n)_{n\in \mathbb {N}}$ in of finite $\mu $ -measure such that $\bigcup _{n\in \mathbb {N}}\Omega _n=\Omega $ . In turn, and $(\forall \omega \in \Omega )\, 1_{\Omega _n}(\omega )\mathsf {x}\to \mathsf {x}$ . Hence, on account of (3.25), we deduce that, for every $n\in \mathbb {N}$ , the mapping $\Omega \to \mathsf {H}\colon \omega \mapsto J_{{\mathsf {A}}_{\omega }}(1_{\Omega _n}(\omega )\mathsf {x})$ is -measurable. In addition, the continuity of the operators $(J_{{\mathsf {A}}_{\omega }})_{\omega \in \Omega }$ yields $(\forall \omega \in \Omega ) J_{{\mathsf {A}}_{\omega }}(1_{\Omega _n}(\omega )\mathsf {x})\to J_{{\mathsf {A}}_{\omega }}\mathsf {x}$ . Altogether, it results from Lemma 2.2(iv) that the mapping $\Omega \to \mathsf {H}\colon \omega \mapsto J_{{\mathsf {A}}_{\omega }}\mathsf {x}$ is -measurable.

(ii) $\Rightarrow $ (i): Applying [Reference Castaing and Valadier14, Lemma III.14] to the mapping $\Omega \times \mathsf {H}\to \mathsf {H}\colon (\omega ,\mathsf {x})\mapsto J_{{\mathsf {A}}_{\omega }}\mathsf {x}$ , we deduce that, for every $x\in \mathfrak {G}$ , the mapping $\omega \mapsto J_{{\mathsf {A}}_{\omega }}(x(\omega ))$ lies in $\mathfrak {G}$ . Therefore, in the setting of (3.23), the family $({\mathsf {A}}_{\omega })_{\omega \in \Omega }$ satisfies Assumption 3.6. Consequently, we conclude via (3.24) and Theorem 3.8(i) that A is maximally monotone.

(ii) $\Leftrightarrow $ (iii): Combine [Reference Attouch1, Lemme 2.1] and [Reference Attouch1, Théorème 2.1].

Proof (i): Let be a dense subset of . On the one hand, property [A] in Definition 1.1 ensures that, for every $n\in \mathbb {N}$ , the function is -measurable. On the other hand, thanks to the continuity of the operators $({\mathsf {L}}_{\omega })_{\omega \in \Omega }$ ,

(3.28)

Altogether, the function is -measurable.

(ii): For every $\mathsf {z}\in \mathsf {G}$ , we deduce from (3.26) that

(3.29)

and, in turn, from (1.6) that $\mathfrak {e}_{\mathsf {L}}\mathsf {z}\in \mathfrak {H}$ . This confirms that L is well defined. In addition, the linearity of the operators $({\mathsf {L}}_{\omega })_{\omega \in \Omega }$ guarantees that of L. The last claims follow from (3.29) and (1.7).

(iii): For every $\mathsf {z}\in \mathsf {G}$ , Lemma 2.2(i) implies that the function is -measurable. In turn, invoking the separabi lity of $\mathsf {G}$ , as well as the fact that is complete and $\sigma $ -finite, we derive from [Reference Schwartz36, Théorème 5.6.24] that the mapping $\Omega \to \mathsf {G}\colon \omega \mapsto {\mathsf {L}}_{\omega }^{*}(x^{*}(\omega ))$ is -measurable.

(iv): By the Cauchy–Schwarz inequality,

(3.30)

Hence, the assertion follows from [Reference Schwartz36, Théorème 5.7.21].

(v): Take $x^{*}\in \mathcal {H}$ . It results from (1.7), (3.27), (3.26), (iv), and [Reference Schwartz36, Théorème 5.8.16] that

(3.31)

which completes the proof.

Proof (i): The core of our argument is implicitly in [Reference Moreau28, Corollaire 10.c]. Suppose that there exists $f\in \Gamma _0(\mathcal {H})$ such that $T=\operatorname { {prox}}_f$ . Then, on account of [Reference Moreau28, Corollaire 10.c] and [Reference Bauschke and Combettes3, Proposition 22.14], T is nonexpansive and cyclically monotone. Conversely, suppose that T is nonexpansive and cyclically monotone. Then T is monotone and it thus follows from [Reference Bauschke and Combettes3, Corollary 20.28] that T is maximally monotone. Therefore, Rockafellar’s cyclic monotonicity theorem [Reference Bauschke and Combettes3, Theorem 22.18] guarantees the existence of a function $\varphi \in \Gamma _0(\mathcal {H})$ such that $T=\partial \varphi $ . We conclude by invoking [Reference Moreau28, Corollaire 10.c].

(ii): See [Reference Zarantonello38, Theorem 1.1].

Proof Set . Then, on account of Proposition 3.4(i), $T\colon \mathcal {H}\to \mathcal {H}$ is nonexpansive. Next, items (ii) and (v) of Proposition 3.12 ensure that the operator $L\colon \mathsf {G}\to \mathcal {H}\colon \mathsf {z}\mapsto \mathfrak {e}_{\mathsf {L}}\mathsf {z}$ is well defined, linear, and bounded, with , and its adjoint is given by

(4.4) $$ \begin{align} L^{*}\colon\mathcal{H}\to\mathsf{G}\colon x^{*}\mapsto \int_{\Omega}{\mathsf{L}}_{\omega}^{*}({x^{*}(\omega)})\mu(d\omega). \end{align} $$

Hence, $L^{*}\circ T\circ L\colon \mathsf {G}\to \mathsf {G}$ is nonexpansive and

(4.5) $$ \begin{align} (\forall\mathsf{z}\in\mathsf{G})\quad L^{*}\big({T(L\mathsf{z})}\big)=\int_{\Omega} {\mathsf{L}}_{\omega}^{*}\big({\operatorname{{prox}}_{{\mathsf{f}}_{\omega}}({\mathsf{L}}_{\omega}\mathsf{z})}\big) \mu(d\omega). \end{align} $$

Therefore, in the light of Lemma 4.1(i), it remains to show that $L^{*}\circ T\circ L$ is cyclically monotone. Toward this end, let $2\leqslant n\in \mathbb {N}$ and let $({\mathsf {z}}_1,\ldots ,{\mathsf {z}}_{n+1})\in {\mathsf {G}}^{n+1}$ be such that ${\mathsf {z}}_{n+1}={\mathsf {z}}_1$ . Then, appealing to the cyclic monotonicity of the operators $(\operatorname { {prox}}_{{\mathsf {f}}_{\omega }})_{\omega \in \Omega }$ ,

(4.6)

Thus, it follows from (1.7) that

(4.7)

which concludes the proof.

Proof Lemma 2.2(v) asserts that there exists a sequence $(u_n)_{n\in \mathbb {N}}$ in $\mathfrak {H}$ such that

(4.9)

(i): Set $\mathbb {I}=\bigl \{{(i_k)_{1\leqslant k\leqslant n+1}\in \mathbb {N}^{n+1}}\mid { 2\leqslant n\in \mathbb {N}\,\,\text {and}\,\,i_{n+1}=i_1}\bigr \}$ , fix temporarily $\boldsymbol {\mathrm {i}}=(i_k)_{1\leqslant k\leqslant n+1}\in \mathbb {I}$ , and let be such that $\mu (\Theta )<{+}\infty $ . Then, by Lemma 2.2(iii), . In turn, since $J_A\colon \mathcal {H}\to \mathcal {H}$ , it follows from Proposition 3.5 and Lemma 2.2(ii) that, for every , a representative in $\mathfrak {H}$ of $J_A(1_{\Theta }u_{i_k})$ is the mapping

(4.10) $$ \begin{align} \omega\mapsto \begin{cases} J_{{\mathsf{A}}_{\omega}}\big({u_{i_k}(\omega)}\big), &\text{if}\,\,\omega\in\Theta,\\ J_{{\mathsf{A}}_{\omega}}\mathsf{0},&\text{if}\,\,\omega\in\complement\Theta. \end{cases} \end{align} $$

At the same time, for every , a representative in $\mathfrak {H}$ of $1_{\Theta }u_{i_k}$ is the mapping

(4.11) $$ \begin{align} \omega\mapsto \begin{cases} u_{i_k}(\omega),&\text{if}\,\,\omega\in\Theta,\\ \mathsf{0},&\text{if}\,\,\omega\in\complement\Theta. \end{cases} \end{align} $$

Hence, since $J_A=\operatorname { {prox}}_f$ is cyclically monotone by virtue of [Reference Bauschke and Combettes3, Example 23.3] and Lemma 4.1, we derive from (1.7) that

(4.12)

Therefore, thanks to the fact that is $\sigma $ -finite, there exists such that

(4.13)

Now set $\Xi =\bigcup _{\boldsymbol {\mathrm {i}}\in \mathbb {I}} \Xi _{\boldsymbol {\mathrm {i}}}$ . Since $\mathbb {I}$ is countable, and $\mu (\Xi )=0$ . Additionally, (4.13) implies that

(4.14)

To proceed further, take $\omega \in \complement\!\Xi $ , let $2\leqslant n\in \mathbb {N}$ , and let $({\mathsf {x}}_1,\ldots ,{\mathsf {x}}_{n+1})$ be a family in ${\mathsf {H}}_{\omega }$ such that ${\mathsf {x}}_{n+1}={\mathsf {x}}_1$ . For every , we infer from (4.9) that there exists a sequence $(i_{k,m})_{m\in \mathbb {N}}$ in $\mathbb {N}$ such that $u_{i_{k,m}}(\omega )\to {\mathsf {x}}_k$ . Set $(\forall m\in \mathbb {N}) i_{n+1,m}=i_{1,m}$ . Then, for every $m\in \mathbb {N}$ , because $(i_{k,m})_{1\leqslant k\leqslant n+1}\in \mathbb {I}$ , it results from (4.14) that

(4.15)

Therefore, the continuity of $J_{{\mathsf {A}}_{\omega }}$ forces . Consequently, since $J_{{\mathsf {A}}_{\omega }}$ is nonexpansive, we conclude via Lemma 4.1(i) that there exists ${\mathsf {f}}_{\omega }\in \Gamma _0({\mathsf {H}}_{\omega })$ such that $J_{{\mathsf {A}}_{\omega }}=\operatorname { {prox}}_{{\mathsf {f}}_{\omega }}$ .

(ii): Argue as in (i).

Proof According to (4.16), there exists such that

(4.18) $$ \begin{align} \mu(\Xi)=0 \end{align} $$

and

(4.19)

Let us define

(4.20) $$ \begin{align} p\colon\omega\mapsto \operatorname{{prox}}_{{\mathsf{f}}_{\omega}}\big({r(\omega)+s^{*}(\omega)}\big). \end{align} $$

Since $r+s^{*}\in \mathfrak {H}$ , Assumption 4.6[B] ensures that $p\in \mathfrak {G}$ . In addition, we deduce from [Reference Bauschke and Combettes3, Proposition 16.44] that

(4.21) $$ \begin{align} \big({\forall\omega\in\Omega}\big)\quad r(\omega)+s^{*}(\omega)-p(\omega)\in \partial{\mathsf{f}}_{\omega}\big({p(\omega)}\big) \end{align} $$

and, in turn, from (2.2) and (4.19) that

(4.22)

On the other hand, thanks to items [C] and [D] in Assumption 4.6, the function lies in . Therefore, it results from (4.22) that $r-p\in \mathfrak {H}$ and, since $r\in \mathfrak {H}$ by Assumption 4.6[C], we get

(4.23) $$ \begin{align} p\in\mathfrak{H}. \end{align} $$

Now set

(4.24)

Assumption 4.6[A] and [Reference Moreau28, Proposition 12.b] imply that the operators $(\partial {\mathsf {f}}_{\omega })_{\omega \in \Omega }$ are maximally monotone. Moreover, since $r+s^{*}\in \mathfrak {H}$ , we infer from (4.21) and (4.23) that $p\in \operatorname { {dom}} A$ . Moreover, Assumption 4.6[B] and [Reference Bauschke and Combettes3, Example 23.3] guarantee that, for every $x\in \mathfrak {H}$ , the mapping $\omega \mapsto J_{\partial {\mathsf {f}}_{\omega }}(x(\omega ))$ lies in $\mathfrak {G}$ . Altogether,

(4.25) $$ \begin{align} \text{the family } (\partial{\mathsf{f}}_{\omega})_{\omega\in\Omega} \text{ satisfies the assumption of Theorem~3.8}. \end{align} $$

Hence, it follows from Theorem 3.8(i) that

(4.26) $$ \begin{align} A \text{ is maximally monotone} \end{align} $$

and from Theorem 3.8(ii)(a) and [Reference Bauschke and Combettes3, Example 23.3] that

(4.27) $$ \begin{align} \big({\forall\gamma\in]{0},{{+}\infty}}[\big)(\forall x\in\mathfrak{H})\quad \text{the mapping}\,\, \omega\mapsto\operatorname{{prox}}_{\gamma{\mathsf{f}}_{\omega}}\big({x(\omega)}\big) \,\,\text{lies in}\,\,\mathfrak{H}. \end{align} $$

(i): We must show that, for every $x\in \mathfrak {H}$ , the function $\Omega \to ]{{-}\infty },{{+}\infty }]\colon \omega \mapsto {\mathsf {f}}_{\omega }(x(\omega ))$ is -measurable. To do so, we employ a Moreau envelope approximation technique inspired by [Reference Attouch1]. Take $x\in \mathfrak {H}$ . For every $\gamma \in ]{0},{{+}\infty }[$ , let $\Psi _{\gamma }$ be the mapping defined on $ [{0},{1}]\times \Omega $ by

(4.28) $$ \begin{align} &\big({\forall(t,\omega)\in[{0},{1}]\times\Omega}\big)\quad \Psi_{\gamma}(t,\omega)\nonumber\\& \quad =r(\omega)+t\big({x(\omega)-r(\omega)}\big) -\operatorname{{prox}}_{\gamma{\mathsf{f}}_{\omega}}\big({ r(\omega)+t\big({x(\omega)-r(\omega)}\big)}\big) \end{align} $$

and define

(4.29)

Then, for every $\gamma \in ]{0},{{+}\infty }[$ , the continuity of the mappings $(\Psi _{\gamma }({\mkern 1.6mu\cdot \mkern 1.6mu},\omega ))_{\omega \in \Omega }$ ensures that the functions $(\phi _{\gamma }({\mkern 1.6mu\cdot \mkern 1.6mu},\omega ))_{\omega \in \Omega }$ are continuous, while (4.27) and Lemma 2.2(i) ensure that the functions $(\phi _{\gamma }(t,{\mkern 1.6mu\cdot \mkern 1.6mu}))_{t\in [{0},{1}]}$ are -measurable. Hence, the functions $(\phi _{\gamma })_{\gamma \in ]{0},{{+}\infty[ }}$ are -measurable [Reference Castaing and Valadier14, Lemma III.14]. In turn, invoking the fact that is $\sigma $ -finite, we deduce that, for every $\gamma \in ]{0},{{+}\infty }[$ , the function $\Omega \to \mathbb {R}\colon \omega \mapsto \int _0^1\phi _{\gamma }(t,\omega )dt$ is -measurable. Therefore, for every $\gamma \in ]{0},{{+}\infty }[$ , since [Reference Bauschke and Combettes3, Proposition 12.30] implies that

(4.30)

we infer that the function is -measurable. However, [Reference Bauschke and Combettes3, Proposition 12.33(ii)] and Assumption 4.6[C] give

(4.31)

Hence, the function $\Omega \to ]{{-}\infty },{{+}\infty }]\colon \omega \mapsto {\mathsf {f}}_{\omega }(x(\omega ))- {\mathsf {f}}_{\omega }(r(\omega ))$ is -measurable. Consequently, invoking Assumption 4.6[C] once more, we conclude that the function $\Omega \to ]{{-}\infty },{{+}\infty }]\colon \omega \mapsto {\mathsf {f}}_{\omega }(x(\omega ))$ is -measurable.

(ii): By (4.19), (4.18), and Assumption 4.6[D],

(4.32)

which yields

(4.33) $$ \begin{align} {-}\infty\notin f(\mathcal{H}). \end{align} $$

At the same time, since the functions $({\mathsf {f}}_{\omega })_{\omega \in \Omega }$ are convex by Assumption 4.6[A], so is f. Moreover, Assumption 4.6[C] implies that $\operatorname { {dom}} f\neq \varnothing $ . Therefore, it remains to show that f is lower semicontinuous. Take $\xi \in \mathbb {R}$ , let $(x_n)_{n\in \mathbb {N}}$ be a sequence in $\mathcal {H}$ , let $x\in \mathcal {H}$ , and suppose that

(4.34) $$ \begin{align} \sup_{n\in\mathbb{N}}f(x_n)\leqslant\xi\; \quad\text{and}\quad x_n\to x. \end{align} $$

Then Lemma 2.3 asserts that there exists a strictly increasing sequence $(k_n)_{n\in \mathbb {N}}$ in $\mathbb {N}$ such that $(\forall ^{\mu }\omega \in \Omega ) x_{k_n}(\omega )\to x(\omega )$ . Let us define

(4.35)

By (i) and Lemma 2.2(i), the functions $(\varrho _n)_{n\in \mathbb {N}}$ are -measurable. Additionally,

(4.36)

and, since the functions $({\mathsf {f}}_{\omega })_{\omega \in \Omega }$ are lower semicontinuous, . Thus, we derive from Fatou’s lemma and (4.34) that

(4.37)

Hence, $f(x)\leqslant \xi $ and we conclude via [Reference Bauschke and Combettes3, Lemma 1.24] that f is lower semicontinuous.

(iii): Let $(x,x^{*})\in \operatorname { {gra}} A$ and let be such that $\mu (\Theta )=0$ and $(\forall \omega \in \complement \Theta ) x^{*}(\omega )\in \partial {\mathsf {f}}_{\omega }(x(\omega ))$ . For every $y\in \mathcal {H}$ , thanks to the inequalities

(4.38)

we obtain . Hence, $(x,x^{*})\in \operatorname { {gra}}\partial f$ and we thus have $\operatorname { {gra}} A\subset \operatorname { {gra}}\partial f$ . Consequently, the monotonicity of $\partial f$ and (4.26) force $\partial f=A$ .

(iv): Combine (ii), [Reference Bauschke and Combettes3, Example 23.3], (iii), (4.25), and Theorem 3.8(ii)(a).

(v): We derive from (ii), [Reference Bauschke and Combettes3, Proposition 16.38], (iii), (4.25), and Theorem 3.8(iii) that

(4.39)

This shows that is a closed subset of $\mathcal {H}$ . On the other hand, for every $x\in \operatorname { {dom}} f$ , it results from Definition 1.4 that, for $\mu $ -almost every $\omega \in \Omega $ , $x(\omega )\in \operatorname { {dom}}{\mathsf {f}}_{\omega }$ and, therefore, that . Consequently,

(4.40)

which yields the desired identities.

(vi): This follows from Fermat’s rule, (iii), and Proposition 3.2(iii).

(vii): Appealing to (4.25), we deduce from Theorem 3.8(v)(a) and [Reference Bauschke and Combettes3, Proposition 17.31(i)] that, for every $x\in \mathfrak {H}$ , the mapping lies in $\mathfrak {G}$ . In addition, by (iii),

(4.41)

Furthermore, for every $\omega \in \Omega $ , [Reference Bauschke and Combettes3, Corollary 17.40] asserts that $\nabla {\mathsf {f}}_{\omega }\colon {\mathsf {H}}_{\omega }\to {\mathsf {H}}_{\omega }$ is strong-to-weak continuous. Consequently, in the light of [Reference Bauschke and Combettes3, Proposition 17.41], the assertion follows from Proposition 3.4(i) applied to the operators $(\nabla {\mathsf {f}}_{\omega })_{\omega \in \Omega }$ .

(viii): Take $x\in \mathfrak {H}$ and define $q\colon \omega \mapsto \operatorname { {prox}}_{\gamma {\mathsf {f}}_{\omega }}(x(\omega ))$ . Then (iv) asserts that $q\in \mathfrak {H}$ and $q=\operatorname { {prox}}_{\gamma f}x$ . Hence, we derive from (ii), [Reference Bauschke and Combettes3, Remark 12.24], and Definition 1.4 that

(4.42)

as claimed.

(ix): Let $x^{*}\in \mathfrak {H}$ , let $(\gamma _n)_{n\in \mathbb {N}}$ be a sequence in $]{0},{1}[$ such that $\gamma _n\downarrow 0$ , and define

(4.43)

For every $n\in \mathbb {N}$ , since Moreau’s decomposition theorem [Reference Bauschke and Combettes3, Theorem 14.3(i)] gives

(4.44)

it follows from (viii) that . Further, we deduce from [Reference Bauschke and Combettes3, Proposition 12.33(ii)] that

(4.45) $$ \begin{align} (\forall\omega\in\Omega)\quad \big({\vartheta_n(\omega)}\big)_{n\in\mathbb{N}}\,\, \text{is increasing and}\,\, \vartheta_n(\omega)\uparrow {\mathsf{f}}_{\omega}^{*}\big({x^{*}(\omega)}\big) \end{align} $$

and, therefore, that the function $\Omega \to ]{{-}\infty },{{+}\infty }]\colon \omega \mapsto {\mathsf {f}}_{\omega }^{*}(x^{*}(\omega ))$ is -measurable. On the other hand, invoking (4.44), (viii), Moreau’s decomposition theorem, and [Reference Bauschke and Combettes3, Proposition 12.33(ii)], we obtain

(4.46)

Thus, in view of (4.45), we infer from the Beppo Levi monotone convergence theorem [Reference Bogachev5, Theorem 2.8.2 and Corollary 2.8.6] that

(4.47) $$ \begin{align} f^{*}(x^{*}) =\lim\int_{\Omega}\vartheta_n(\omega)\mu(d\omega) =\int_{\Omega}\lim\vartheta_n(\omega)\,\mu(d\omega) =\int_{\Omega} {\mathsf{f}}_{\omega}^{*}\big({x^{*}(\omega)}\big)\mu(d\omega). \end{align} $$

(x): Assumption 4.6[C] ensures that $(\forall \omega \in \Omega ) r(\omega )\in \operatorname { {dom}}{\mathsf {f}}_{\omega }$ . Now take $x\in \mathcal {H}$ and set

(4.48) $$ \begin{align} \big({\forall\alpha\in]{0},{{+}\infty}}[\big)\quad \theta_{\alpha}\colon\Omega\to]{{-}\infty},{{+}\infty}]\colon\omega\mapsto \frac{{\mathsf{f}}_{\omega}\big({r(\omega)+\alpha x(\omega)}\big)-{\mathsf{f}}_{\omega}\big({r(\omega)}\big)}{\alpha}. \end{align} $$

Then, for every $\alpha \in ]{0},{{+}\infty }[$ , since $r+\alpha x\in \mathfrak {H}$ and $r\in \mathfrak {H}$ , it results from (i) that $\theta _{\alpha }$ is -measurable. On the other hand, by Assumption 4.6[A] and [Reference Bauschke and Combettes3, Propositions 9.27 and 9.30(ii)], we obtain

(4.49) $$ \begin{align} &(\forall\omega\in\Omega)\quad \text{the net}\,\,\big({\theta_{\alpha}(\omega)}\big)_{\alpha\in]{0},{{+}\infty}[} \,\,\text{is increasing}\nonumber\\& \quad \text{and}\,\, (\operatorname{{rec}}{\mathsf{f}}_{\omega})\big({x(\omega)}\big) =\lim_{\alpha\uparrow{+}\infty}\theta_{\alpha}(\omega). \end{align} $$

Altogether, we infer from the Beppo Levi monotone convergence theorem, Assumption 4.6[C], (ii), and [Reference Bauschke and Combettes3, Proposition 9.30(ii)] that

(4.50) $$ \begin{align} \int_{\Omega}(\operatorname{{rec}}{\mathsf{f}}_{\omega})\big({x(\omega)}\big) \mu(d\omega) &=\int_{\Omega}\lim_{\alpha\uparrow{+}\infty} \theta_{\alpha}(\omega)\,\mu(d\omega) \nonumber\\ &=\lim_{\alpha\uparrow{+}\infty}\int_{\Omega} \theta_{\alpha}(\omega)\mu(d\omega) \nonumber\\ &=\lim_{\alpha\uparrow{+}\infty}\frac{1}{\alpha}\left({ \int_{\Omega}{\mathsf{f}}_{\omega}\big({r(\omega)+\alpha x(\omega)}\big)\mu(d\omega)- \int_{\Omega}{\mathsf{f}}_{\omega}\big({r(\omega)}\big)\mu(d\omega)}\right) \nonumber\\ &=\lim_{\alpha\uparrow{+}\infty}\frac{f(r+\alpha x)-f(r)}{\alpha} \nonumber\\ &=(\operatorname{{rec}} f)(x), \end{align} $$

which completes the proof.

Proof (ii) $\Rightarrow $ (i): Use Moreau’s theorem [Reference Moreau28, Proposition 12.b].

(i) $\Rightarrow $ (iii): This is a special case of Corollary 3.10.

(iii) $\Rightarrow $ (ii): Set . Then, as seen in Example 2.1(iv), . For every $\omega \in \Omega $ , [Reference Bauschke and Combettes3, Corollary 22.23] asserts that there exists ${\mathsf {g}}_{\omega }\in \Gamma _0(\mathbb {R})$ such that ${\mathsf {A}}_{\omega }=\partial {\mathsf {g}}_{\omega }$ . Next, since $\operatorname { {dom}} A\neq \varnothing $ and is complete, there exist r and $s^{*}$ in such that

(4.53) $$ \begin{align} (\forall^{\mu}\omega\in\Omega)\quad s^{*}(\omega) \in{\mathsf{A}}_{\omega}\big({r(\omega)}\big) =\partial{\mathsf{g}}_{\omega}\big({r(\omega)}\big) \end{align} $$

and

(4.54) $$ \begin{align} (\forall\omega\in\Omega)\quad r(\omega)\in\operatorname{{dom}}{\mathsf{A}}_{\omega}\subset\operatorname{{dom}}{\mathsf{g}}_{\omega}. \end{align} $$

Now set

(4.55) $$ \begin{align} (\forall\omega\in\Omega)\quad {\mathsf{f}}_{\omega} ={\mathsf{g}}_{\omega} -{\mathsf{g}}_{\omega}\big({r(\omega)}\big). \end{align} $$

Then the functions $({\mathsf {f}}_{\omega })_{\omega \in \Omega }$ lie in $\Gamma _0(\mathbb {R})$ and, by [Reference Bauschke and Combettes3, Proposition 24.8(i) and Example 23.3], $(\forall \omega \in \Omega ) \operatorname { {prox}}_{{\mathsf {f}}_{\omega }} =\operatorname { {prox}}_{{\mathsf {g}}_{\omega }} =J_{{\mathsf {A}}_{\omega }}$ . In turn, appealing to the continuity of the operators $(J_{{\mathsf {A}}_{\omega }})_{\omega \in \Omega }$ , we deduce from [Reference Castaing and Valadier14, Lemma III.14] that the mapping $\Omega \times \mathbb {R}\to \mathbb {R}\colon (\omega ,\mathsf {x}) \mapsto \operatorname { {prox}}_{{\mathsf {f}}_{\omega }}\mathsf {x}$ is -measurable. Therefore, for every , the mapping $\Omega \to \mathbb {R}\colon \omega \mapsto \operatorname { {prox}}_{{\mathsf {f}}_{\omega }}(x(\omega ))$ lies in $\mathfrak {G}$ . Next, we get from (4.55) and (4.54) that $(\forall \omega \in \Omega ) {\mathsf {f}}_{\omega }(r(\omega ))=0$ . Moreover, by (4.53),

(4.56) $$ \begin{align} (\forall^{\mu}\omega\in\Omega)(\forall\mathsf{x}\in\mathbb{R})\quad {\mathsf{f}}_{\omega}(\mathsf{x}) &={\mathsf{g}}_{\omega}(\mathsf{x})- {\mathsf{g}}_{\omega}\big({r(\omega)}\big) \geqslant \big({\mathsf{x}-r(\omega)}\big)s^{*}(\omega)\nonumber\\ &=\mathsf{x}s^{*}(\omega)-r(\omega)s^{*}(\omega). \end{align} $$

Hence, since $\omega \mapsto r(\omega )s^{*}(\omega )$ lies in , the family $({\mathsf {f}}_{\omega })_{\omega \in \Omega }$ satisfies the assumption of Theorem 4.7. Altogether, we conclude via Theorem 4.7(ii) that

(4.57)

and via Theorem 4.7(iii) and (4.52) that

(4.58)

as desired.

Proof Apply Corollary 4.10 to the case where $\Omega =\mathbb {N}$ , , and $\mu $ is the counting measure.

Proof Set $(\forall \omega \in \Omega ) {\mathsf {f}}_{\omega }=\iota _{{\mathsf {C}}_{\omega }}$ . Then, for every $\omega \in \Omega $ , ${\mathsf {f}}_{\omega }\in \Gamma _0({\mathsf {H}}_{\omega })$ , ${\mathsf {f}}_{\omega }\geqslant 0$ , and $\operatorname { {prox}}_{{\mathsf {f}}_{\omega }}=\operatorname { {proj}}_{{\mathsf {C}}_{\omega }}$ . Moreover, since $C\neq \varnothing $ and is complete, there exists $r\in \mathfrak {H}$ such that, for every $\omega \in \Omega $ , $r(\omega )\in {\mathsf {C}}_{\omega }$ or, equivalently, ${\mathsf {f}}_{\omega }(r(\omega ))=0$ . Altogether, the family $({\mathsf {f}}_{\omega })_{\omega \in \Omega }$ satisfies the assumption of Theorem 4.7. Therefore, in view of items (i) and (ii) in Theorem 4.7,

(4.61)

(i): Using Definitions 1.4 and 3.1, together with (4.60), we obtain

(4.62) $$ \begin{align} (\forall x\in\mathcal{H})\quad f(x) &=\int_{\Omega}\iota_{{\mathsf{C}}_{\omega}}\big({x(\omega)}\big) \mu(d\omega) \nonumber\\ &= \begin{cases} 0,&\text{if}\,\,(\forall^{\mu}\omega\in\Omega)\,\,x(\omega)\in {\mathsf{C}}_{\omega},\\ {+}\infty,&\text{otherwise,} \end{cases} \nonumber\\ &= \begin{cases} 0,&\text{if}\,\,x\in C,\\ {+}\infty,&\text{otherwise,} \end{cases} \nonumber\\ &=\iota_C(x), \end{align} $$

and the claim thus follows from (4.61).

(ii)–(v): In the light of (4.61) and (4.62), these follow from items (iii), (iv), (viii), and (ix) in Theorem 4.7, respectively.

(vi): We deduce from [Reference Bauschke and Combettes3, Example 6.40] and (ii) that

(4.63)

(vii): Use (vi) and [Reference Bauschke and Combettes3, Proposition 6.23].

Proof Theorem 4.7(i)–(ii) states that

(4.66)

On the other hand, according to Proposition 3.12(ii),

(4.67) $$ \begin{align} L\colon\mathsf{G}\to\mathcal{H}\colon\mathsf{z}\mapsto \mathfrak{e}_{\mathsf{L}}\mathsf{z} \,\,\text{is well defined, linear, and bounded}. \end{align} $$

(i): Because $L\mathsf {w}\in \operatorname { {dom}} f$ , it follows from (4.64), (4.66), and (4.67) that

(4.68) $$ \begin{align} \mathsf{g}=f\circ L\in\Gamma_0(\mathsf{G}). \end{align} $$

(ii): It results from Theorem 4.7(iii), Proposition 3.5, and Moreau’s decomposition [Reference Bauschke and Combettes3, Theorem 14.3(ii)] that

(4.69)

Hence, for every $\gamma \in ]{0},{{+}\infty }[$ and every $\mathsf {w}\in \mathsf {G}$ , since $\mathfrak {e}_{\mathsf {L}}\mathsf {w}\colon \omega \mapsto {\mathsf {L}}_{\omega }\mathsf {w}$ is a representative in $\mathfrak {H}$ of $L\mathsf {w}$ , Proposition 3.12(v) implies that

(4.70)

In addition, appealing to (4.66)–(4.68), we derive from [Reference Pennanen, Revalski and Théra32, Theorem 4.1] and a remark on [Reference Pennanen, Revalski and Théra32, p. 88] that $\operatorname { {gra}}\partial \mathsf {g}$ is the set of points $(\mathsf {w},{\mathsf {w}}^{*})\in \mathsf {G}\times \mathsf {G}$ for which there exist sequences $(\gamma _n)_{n\in \mathbb {N}}$ in $]{0},{{+}\infty }[$ and $({\mathsf {w}}_n)_{n\in \mathbb {N}}$ in $\mathsf {G}$ such that $\gamma _n\downarrow 0$ , ${\mathsf {w}}_n\to \mathsf {w}$ , and . Altogether, the proof is complete.

Proof Suppose that $(\mathsf {z},{\mathsf {z}}^{*})$ and $(\mathsf {w},{\mathsf {w}}^{*})$ are in $\operatorname { {gra}}\mathsf {M}$ . Then, by (2.4), there exist $x^{*}$ and $y^{*}$ in $\prod _{\omega \in \Omega }{\mathsf {H}}_{\omega }$ such that

(5.2)

The monotonicity of the operators $({\mathsf {A}}_{\omega })_{\omega \in \Omega }$ ensures that

(5.3)

Therefore, using [Reference Schwartz36, Théorème 5.8.16], we obtain

(5.4)

which yields the assertion.

Proof Set

(5.10)

Theorem 3.8(i) states that

(5.11) $$ \begin{align} A \text{ is maximally monotone}, \end{align} $$

while Proposition 3.2(iv) states that

(5.12)

Moreover, in view of Assumption 5.2, items (ii) and (v) of Proposition 3.12 imply that the operator

(5.13) $$ \begin{align} L\colon\mathsf{G}\to\mathcal{H}\colon\mathsf{z}\mapsto \mathfrak{e}_{\mathsf{L}}\mathsf{z} \end{align} $$

is well defined, linear, and bounded, with adjoint

(5.14) $$ \begin{align} L^{*}\colon\mathcal{H}\to\mathsf{G}\colon x^{*}\mapsto \int_{\Omega}{\mathsf{L}}_{\omega}^{*}\big({x^{*}(\omega)}\big)\mu(d\omega). \end{align} $$

Hence, we deduce from (5.8) that

(5.15)

and from (5.9) that

(5.16)

Additionally, the dual problem (5.7) can be rewritten as

(5.17) $$ \begin{align} \text{find}\,\,x^{*}\in\mathcal{H}\,\,\text{such that}\,\, 0\in{-}L\big({\mathsf{W}^{-1}(-L^{*}x^{*})}\big)+A^{-1}x^{*}. \end{align} $$

(i): In view of (5.11) and the maximal monotonicity of $\mathsf {W}$ , it follows from (5.15) and [Reference Bauschke and Combettes3, Proposition 26.32(iii)] that is maximally monotone, and from (5.16) and [Reference Bùi and Combettes9, Lemma 2.2(ii)] that is maximally monotone.

(ii): Combine (i) and [Reference Bauschke and Combettes3, Proposition 23.39].

(iii): Suppose that . Then, by (5.15), $L\mathsf {z}\in A^{-1}x^{*}$ or, equivalently, $x^{*}\in A(L\mathsf {z})$ . Therefore, it follows from (5.13), Assumption 5.2[C], and (5.10) that, for $\mu $ -almost every $\omega \in \Omega $ , $x^{*}(\omega )\in {\mathsf {A}}_{\omega }({\mathsf {L}}_{\omega }\mathsf {z})$ and, in turn, that ${\mathsf {L}}_{\omega }^{*}(x^{*}(\omega ))\in {\mathsf {L}}_{\omega }^{*}({\mathsf {A}}_{\omega }({\mathsf {L}}_{\omega }\mathsf {z}))$ . Hence, because Proposition 3.12(iv) asserts that the mapping $\Omega \to \mathsf {G}\colon \omega \mapsto {\mathsf {L}}_{\omega }^{*}(x^{*}(\omega ))$ is $\mu $ -integrable, we infer from (5.8) and (2.4) that

(5.18) $$ \begin{align} \mathsf{0} \in\mathsf{W}\mathsf{z}+ \int_{\Omega}{\mathsf{L}}_{\omega}^{*}\big({x^{*}(\omega)}\big)\mu(d\omega) \subset\mathsf{W}\mathsf{z}+ \int_{\Omega}{\mathsf{L}}_{\omega}^{*}\big({{\mathsf{A}}_{\omega}({\mathsf{L}}_{\omega}\mathsf{z})}\big)\mu(d\omega). \end{align} $$

Finally, since , it follows from [Reference Bauschke and Combettes3, Proposition 26.33(ii)] that $x^{*}$ solves (5.17) and, therefore, (5.7).

(iv): Argue as in (iii).

(v): By virtue of (5.15)–(5.17), the equivalences follow from [Reference Bùi and Combettes9, Lemma 2.2(iv)], while the implication follows from (iii).