Proof of A Conditional Density Identity for Products Extends to Jointly Measurable Integrands

Step 0 (Integrals with respect to the restriction of $P$ to $\mathcal{T}$). Let $P_{\mathcal{T}}$ denote the restriction of $P$ to $\mathcal{T}$; it is a measure on $(\Omega,\mathcal{T})$ with $P_{\mathcal{T}}(\Omega)=1$, countable additivity for members of $\mathcal{T}$ being a special case of that for members of $\mathcal{F}$, so $(\Omega,\mathcal{T},P_{\mathcal{T}})$ is a finite, hence $\sigma$-finite, measure space. We claim that for every $\mathcal{T}$-measurable $\Xi:\Omega\to[0,\infty]$,

$$\int_\Omega\Xi\,dP_{\mathcal{T}}=\int_\Omega\Xi\,dP=\mathbb{E}[\Xi]\qquad\text{in }[0,\infty].\qquad(0)$$

For $m\ge1$ let $s_m=\sum_{p=1}^{m2^m}2^{-m}\mathbf{1}\{\Xi\ge p2^{-m}\}$, where $\mathbf{1}\{\cdots\}$ is the indicator of the event in braces. Each $s_m$ is a nonnegative simple function on $(\Omega,\mathcal{T})$, the sets $\{\Xi\ge p2^{-m}\}=\bigcap_{n}\{\Xi>p2^{-m}-1/n\}$ lying in $\mathcal{T}$, and its integral with respect to $P_{\mathcal{T}}$ and with respect to $P$ is the same number, the integral of a nonnegative simple function being computed from the measures of the level sets of its standard representation, which are members of $\mathcal{T}$ on which $P_{\mathcal{T}}$ and $P$ agree. At every $\omega$: $s_m(\omega)=2^{-m}\min\bigl(m2^m,\lfloor2^m\Xi(\omega)\rfloor\bigr)$ when $\Xi(\omega)<\infty$, with $\lfloor\cdot\rfloor$ the integer part, since $p\le2^m\Xi(\omega)$ holds for exactly $\min(m2^m,\lfloor2^m\Xi(\omega)\rfloor)$ indices $p\in\{1,\dots,m2^m\}$, and $s_m(\omega)=m$ when $\Xi(\omega)=\infty$. Hence the sequence $(s_m(\omega))_m$ is nondecreasing (from $2\lfloor y\rfloor\le\lfloor2y\rfloor$, which holds because $2\lfloor y\rfloor$ is an integer at most $2y$ and every integer $n\le x$ satisfies $n\le\lfloor x\rfloor$, as $n\ge\lfloor x\rfloor+1$ would give $n>x$; and from the growth of the cap $m$) and converges to $\Xi(\omega)$ (for $\Xi(\omega)<\infty$ and $m\ge\Xi(\omega)$ one has $\Xi(\omega)-2^{-m}\le s_m(\omega)\le\Xi(\omega)$; for $\Xi(\omega)=\infty$, $s_m(\omega)=m\to\infty$). The monotone convergence theorem, applied on $(\Omega,\mathcal{T},P_{\mathcal{T}})$ and on $(\Omega,\mathcal{F},P)$ (where $s_m$ and $\Xi$ are also measurable, as $\mathcal{T}\subseteq\mathcal{F}$), gives (0), both sides being the supremum of the same sequence of numbers.

Step 1 (The pairing map). Let $\Phi:\Omega\to\mathbf{R}\times\Omega$, $\Phi(\omega)=(W(\omega),\omega)$. For a measurable rectangle $A\times C$ with $A\in\mathcal{R}$ and $C\in\mathcal{T}$, $\Phi^{-1}(A\times C)=W^{-1}(A)\cap C\in\mathcal{F}$. Since the family of subsets of $\mathbf{R}\times\Omega$ whose preimage under $\Phi$ lies in $\mathcal{F}$ is a $\sigma$-algebra (preimages commute with complements and countable unions) containing the rectangles, which generate $\mathcal{R}\otimes\mathcal{T}$ (Generated Sigma-Algebra), $\Phi$ is measurable with respect to $\mathcal{F}$ and $\mathcal{R}\otimes\mathcal{T}$. Consequently, for $H$ as in the statement, $\omega\mapsto H(W(\omega),\omega)=(H\circ\Phi)(\omega)$ is $\mathcal{F}$-measurable, since $\{H\circ\Phi>a\}=\Phi^{-1}(\{H>a\})$ for every real $a$. Let $P_\Phi$ be the image measure of $P$ under $\Phi$ on $(\mathbf{R}\times\Omega,\mathcal{R}\otimes\mathcal{T})$ (claim 1 of that lemma); it satisfies $P_\Phi(\mathbf{R}\times\Omega)=1$.

Step 2 (The density measure). The product measure $\rho\otimes P_{\mathcal{T}}$ exists on $\mathcal{R}\otimes\mathcal{T}$, both factors being $\sigma$-finite. By the Tonelli theorem, for every $\mathcal{R}\otimes\mathcal{T}$-measurable $\Psi:\mathbf{R}\times\Omega\to[0,\infty]$ the sections $r\mapsto\Psi(r,\omega)$ are $\mathcal{R}$-measurable, the map $\omega\mapsto\int_{\mathbf{R}}\Psi(r,\omega)\,\rho(dr)$ is $\mathcal{T}$-measurable, and

$$\int_{\mathbf{R}\times\Omega}\Psi\,d(\rho\otimes P_{\mathcal{T}})=\int_\Omega\Bigl(\int_{\mathbf{R}}\Psi(r,\omega)\,\rho(dr)\Bigr)P_{\mathcal{T}}(d\omega)=\mathbb{E}\Bigl[\int_{\mathbf{R}}\Psi(r,\cdot)\,\rho(dr)\Bigr],\qquad(1)$$

the last equality by (0). Let $\pi$ be the measure with density $f$ with respect to $\rho\otimes P_{\mathcal{T}}$ (claim 3 of that lemma; $f$ is $\mathcal{R}\otimes\mathcal{T}$-measurable with values in $[0,\infty)$), so that $\pi(S)=\int\mathbf{1}_Sf\,d(\rho\otimes P_{\mathcal{T}})$ for $S\in\mathcal{R}\otimes\mathcal{T}$ and $\int\Psi\,d\pi=\int\Psi f\,d(\rho\otimes P_{\mathcal{T}})$ for every $\mathcal{R}\otimes\mathcal{T}$-measurable $\Psi\ge0$, the product $\Psi f$ being measurable (claim 1 of Linearity and Monotonicity of the Lebesgue Integral covers sums and constant multiples; for the product, $\{\Psi f>a\}=\bigcup_{q}(\{\Psi>q\}\cap\{f>a/q\})$ over positive rationals $q$ when $a\ge0$, and $\{\Psi f>a\}=\mathbf{R}\times\Omega$ when $a<0$).

Step 3 (The two measures agree). For a measurable rectangle $A\times C$ with $A\in\mathcal{R}$ and $C\in\mathcal{T}$, apply (1) to $\Psi=\mathbf{1}_{A\times C}f$, whose section at $\omega$ is $\mathbf{1}_C(\omega)\mathbf{1}_A(r)f(r,\omega)$, the constant factor $\mathbf{1}_C(\omega)$ coming out of the inner integral by claim 1 of Linearity and Monotonicity of the Lebesgue Integral:

$$\pi(A\times C)=\mathbb{E}\Bigl[\mathbf{1}_C\int_{\mathbf{R}}\mathbf{1}_A(r)f(r,\cdot)\,\rho(dr)\Bigr]=\mathbb{E}\bigl[\mathbf{1}_C\,\mathbf{1}_A(W)\bigr]=P\bigl(C\cap W^{-1}(A)\bigr)=P_\Phi(A\times C),$$

the second equality being the hypothesis for $C$ and $A$, and the third the integral of the simple function $\mathbf{1}_{C\cap W^{-1}(A)}$. In particular, with $A=\mathbf{R}$ and $C=\Omega$, $\pi(\mathbf{R}\times\Omega)=1=P_\Phi(\mathbf{R}\times\Omega)$. The measurable rectangles form a $\pi$-system (the intersection of $A\times C$ and $A'\times C'$ is $(A\cap A')\times(C\cap C')$) generating $\mathcal{R}\otimes\mathcal{T}$, and $\pi$ and $P_\Phi$ are measures on $\mathcal{R}\otimes\mathcal{T}$ of equal finite total mass agreeing on it; by claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, $\pi=P_\Phi$.

Step 4 (Conclusion). Let $H:\mathbf{R}\times\Omega\to[0,\infty]$ be $\mathcal{R}\otimes\mathcal{T}$-measurable. The $\mathcal{T}$-measurability of $\omega\mapsto\int_{\mathbf{R}}H(r,\omega)f(r,\omega)\,\rho(dr)$ and the $\mathcal{R}$-measurability of the sections are the Tonelli assertions of Step 2 applied to $\Psi=Hf$. Finally, by claim 2 of Image Measures, Measures with Densities, and Change of Variables (change of variables for $P_\Phi$), Step 3, claim 3 of that lemma (the density $f$ of $\pi$), and (1) applied to $\Psi=Hf$,

$$\mathbb{E}\bigl[H(W,\cdot)\bigr]=\int_\Omega H\circ\Phi\,dP=\int_{\mathbf{R}\times\Omega}H\,dP_\Phi=\int_{\mathbf{R}\times\Omega}H\,d\pi=\int_{\mathbf{R}\times\Omega}Hf\,d(\rho\otimes P_{\mathcal{T}})=\mathbb{E}\Bigl[\int_{\mathbf{R}}H(r,\cdot)f(r,\cdot)\,\rho(dr)\Bigr],$$

which is the asserted identity.