Let $\tilde{D}$ be a topological space with Borel $\sigma$-algebra $\mathcal{B}(\tilde{D})$,
$\tilde{g}: \Omega \times \tilde{D} \rightarrow \mathbb{R}$ be a bounded $(\mathcal{G} \otimes \mathcal{B}(\tilde{D}) ) / \mathcal{B}(\mathbb{R})$-measurable function,
$\tilde{\Pi}: \Omega \rightarrow \tilde{D}$ be a $\mathcal{G}/\mathcal{B}(\tilde{D})$-measurable random variable,
$\tilde{Y} : \Omega \rightarrow \mathbb{R}$ be a $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable random variable, defined by $\tilde{Y}(\omega) := \tilde{g}(\omega, \tilde{\Pi}(\omega))$,
$\tilde{s}: \tilde{D} \rightarrow \mathcal{L}^2(\Omega, \mathbb{R}, \mathbb{P}) $ be defined by $\tilde{s}(\pi)(\omega) := \tilde{g}(\omega, \pi)$,
$\tilde{E} : \tilde{D} \rightarrow \mathbb{R}$ be defined by $\tilde{E}(\pi) = \mathbb{E}[\tilde{s}(\pi)]$, and
for all $\pi \in \tilde{D}$, let $\tilde{s}(\pi)$ be independent of $\tilde{\Pi}$.
In this setting, I want to prove the following statement:
$\tilde{E}$ is $\mathcal{B}(\tilde{D})/\mathcal{B}(\mathbb{R})$-measurable and $ \tilde{E} \circ \tilde{\Pi}$ is a version of the $\mathbb{P}$-unique conditional expectation $\mathbb{E}[\tilde{Y} \mid \tilde{\Pi}]$.
Is this even true? An idea for a proof would be fantastic!