# Conditional Expectation of Composite Function

Preliminaries

Let $$(\Omega, \mathcal{G}, \mathbb{P})$$ be a complete probability space.

Let $$D$$ be a complete, separable, metrizable topological space with Borel $$\sigma$$-algebra $$\mathcal{B}(D)$$ (such as $$D = \mathbb{R}^q$$ with $$\sigma$$-algebra $$\mathcal{B}(D) = \mathcal{B}(\mathbb{R}^d)$$).

Let $$\mathbb{R}$$ be equipped with its canonical Borel $$\sigma$$-algebra $$\mathcal{B}(\mathbb{R})$$.

Let $$g: \Omega \times D \rightarrow \mathbb{R}$$ be a bounded $$(\mathcal{G} \otimes \mathcal{B}(D) ) / \mathcal{B}(\mathbb{R})$$-measurable function.

Let $$\Pi: \Omega \rightarrow D$$ be a $$\mathcal{G}/\mathcal{B}(D)$$-measurable random variable.

Let $$H : \Omega \rightarrow \mathbb{R}$$ be a $$\mathcal{G}/\mathcal{B}(\mathbb{R})$$-measurable random variable, defined by $$H(\omega) := g(\omega, \Pi(\omega)).$$ Note, that, since $$g$$ is bounded, we have $$H \in \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P})$$.

Let $$j: D \rightarrow \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P})$$ be defined by $$j(\pi)(\omega) := g(\omega, \pi)$$

For all $$\pi \in D$$, let $$j(\pi)$$ be independent of $$\Pi$$.

Question

I am interested in the conditional expectation $$\mathbb{E}[H \mid \Pi] :\Omega \rightarrow \mathbb{R}$$ of $$H$$ with respect to $$\Pi$$. More specifically, I suspect that (a $$\mathbb{P}$$-unique version of) this condititional expectation is given by

$$\mathbb{E}[H \mid \Pi] (\omega) = \mathbb{E}[j(\Pi(\omega))], \quad (\dagger)$$ whereby $$\mathbb{E}[j(\Pi(\omega))]$$ can of course also be written as $$\mathbb{E}[j(\Pi(\omega))] = \int_{\Omega} j(\Pi(\omega))(\tilde{\omega}) d\mathbb{P}(\tilde{\omega}) .$$

How can I prove, that $$(\dagger)$$ is the case? I have tried, tracking the definition of conditional expectation and using Fubini, but with little success so far.