I apologise upfront for imprecisions of the question. I'm new to measure theory.

$(\mathcal{X}, B_{\mathcal{X}})$ is a measurable space, where $\mathcal{X}\subseteq\mathbb{R}^L$. Denote by $Q_{\mathcal{X}}$ a set of probability measures over $(\mathcal{X}, B_{\mathcal{X}})$.

$(\mathcal{M}, B_{\mathcal{M}}, \sigma)$ is a probability space. [If necessary, assume $\mathcal{M} \subseteq \mathbb{R}^K$.]

Consider a mapping $\mu:\mathcal{M} \rightarrow Q_{\mathcal{X}}$. This mapping selects a probability measure $\mu_m$ over $(\mathcal{X}, B_{\mathcal{X}})$ for each $m\in\mathcal{M}$. Hence, $(\mathcal{X}, B_{\mathcal{X}}, \mu_m)$ is a probability space for each $m\in\mathcal{M}$.

Define a probability measure $e_{\mu}$ over $(\mathcal{X}, B_{\mathcal{X}})$ in the following way \begin{equation} e_{\mu}(X) \equiv \int_{m\in\mathcal{M}}\mu_m(X)\sigma(dm)\ \ \forall X\in B_{\mathcal{X}}. \end{equation}

The idea is that $e_{\mu}$ is a lottery over $\mathcal{X}$ which simulates the outcomes of the lottery $\sigma$ of lotteries $\{\mu_m\}_{m\in\mathcal{M}}$ over $\mathcal{X}$.

Question: Take a continuous function $g:\mathcal{X} \rightarrow \mathbb{R}_+$. Is the following equality always true? \begin{equation} \int_{m\in\mathcal{M}} \int_{x\in\mathcal{X}}g(x)\mu_m(dx)\sigma(dm) = \int_{x\in\mathcal{X}}g(x)e_{\mu}(dx) \end{equation}

In words: can we always combine "a lottery of lotteries" into a single lottery?


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