Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a (complete) probability space and $D$ be a compact topological space, equipped with its canonical Borel $\sigma$-algebra $\mathcal{B}(D)$.

Furthermore, let $$ \Pi : \Omega \rightarrow D$$ be a $\mathcal{G}/\mathcal{B}(D)$-measurable random variable and $$g : \Omega \times D \to \mathbb{R} $$ be a bounded and $(\mathcal{G} \otimes \mathcal{B}(D))/\mathcal{B}(\mathbb{R})$-measurable function, such that, for each fixed $\pi \in D$, the random variable $g(\cdot, \pi)$ is independent of $\Pi$.

Now let $A \in \sigma(\Pi) $. I want to show, that

$$\int_{\Omega} \int_{\Omega} 1_{A}(\omega) g(\tilde{\omega}, \Pi(\omega)) d\mathbb{P}(\tilde{\omega}) d\mathbb{P}(\omega) = \int_{\Omega} 1_{A}(\omega) g(\omega, \Pi(\omega)) d\mathbb{P}(\omega). $$

I am convinced, that under our assumptions, this should be the case, but I am completely stuck. I would be extremely grateful for any advice!


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