A Conditional Density Identity for Products Extends to Jointly Measurable Integrands

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{T}\subseteq\mathcal{F}$ be a sub-$\sigma$-algebra, let $(\mathbf{R},\mathcal{R},\rho)$ be a $\sigma$-finite measure space, and let $W:\Omega\to\mathbf{R}$ be measurable with respect to $\mathcal{F}$ and $\mathcal{R}$. Write $\mathcal{R}\otimes\mathcal{T}$ for the product $\sigma$-algebra on $\mathbf{R}\times\Omega$; measurability of $[0,\infty]$-valued maps and their integrals are those of Lebesgue Integral of a Nonnegative Measurable Function, pointwise products of $[0,\infty]$-valued maps being formed with the multiplication conventions ($0\cdot\infty=0$, $a\cdot\infty=\infty$ for $a>0$) of Image Measures, Measures with Densities, and Change of Variables, and $\mathbb{E}$ denotes the integral with respect to $P$ of a $[0,\infty]$-valued $\mathcal{F}$-measurable map, extending the expectation of a nonnegative random variable. Let $f:\mathbf{R}\times\Omega\to[0,\infty)$ be measurable with respect to $\mathcal{R}\otimes\mathcal{T}$ and suppose that for every $C\in\mathcal{T}$ and every $A\in\mathcal{R}$,

$$\mathbb{E}\bigl[\mathbf{1}_C\,\mathbf{1}_A(W)\bigr]=\mathbb{E}\Bigl[\mathbf{1}_C\int_{\mathbf{R}}\mathbf{1}_A(r)\,f(r,\cdot)\,\rho(dr)\Bigr]\qquad\text{in }[0,\infty],$$

where $\mathbf{1}_S$ denotes the indicator of a set $S$ and where, for each $\omega$, the inner integral is that of the $\mathcal{R}$-measurable section $r\mapsto\mathbf{1}_A(r)f(r,\omega)$, the map $\omega\mapsto\int_{\mathbf{R}}\mathbf{1}_A(r)f(r,\omega)\,\rho(dr)$ being $\mathcal{T}$-measurable by the Tonelli theorem applied on $\mathbf{R}\times\Omega$ with the restriction of $P$ to $\mathcal{T}$. (This hypothesis is the special case $Z=\mathbf{1}_C$, $g=\mathbf{1}_A$ of the identity $\mathbb{E}[Zg(W)]=\mathbb{E}[Z\int_{\mathbf{R}}g(r)f(r,\cdot)\,\rho(dr)]$ for arbitrary $\mathcal{T}$-measurable $Z:\Omega\to[0,\infty]$ and $\mathcal{R}$-measurable $g:\mathbf{R}\to[0,\infty]$, so it holds whenever that identity does.)

Then for every $\mathcal{R}\otimes\mathcal{T}$-measurable $H:\mathbf{R}\times\Omega\to[0,\infty]$: the map $\omega\mapsto H(W(\omega),\omega)$ is $\mathcal{F}$-measurable; the map $\omega\mapsto\int_{\mathbf{R}}H(r,\omega)f(r,\omega)\,\rho(dr)$ is $\mathcal{T}$-measurable, each inner integral being that of an $\mathcal{R}$-measurable section; and

$$\mathbb{E}\bigl[H(W,\cdot)\bigr]=\mathbb{E}\Bigl[\int_{\mathbf{R}}H(r,\cdot)\,f(r,\cdot)\,\rho(dr)\Bigr]\qquad\text{in }[0,\infty].$$