Let $ (\Omega, \mathcal{F} ),P $ be a probability space, such that $(\Omega, \mathcal{F})$ is a Polish space. Let $(S, \mathcal{S}) $ be a Polish space. Let $ X : \Omega \rightarrow S$ be a measurable function. Let $\mathcal{G} $ be a sub $\sigma$-algebra of $\mathcal{F}$. Let $\mu$ be a regular conditional distribution of $X$ give $\mathcal{G}$, i.e. \begin{equation} \mu = E\left[ 1( X \in \cdot ) \mid \mathcal{G} \right]. \end{equation} Let $M_1$ be the space of probability measures on $(S, \mathcal{S})$. Then $\mu$ is a map from $\Omega$ to $M_1$. Let $\mathcal{M}$ be the sigma algebra on $M_1$ generated by the topology of weak convergence.

Is $\mu$ a $\mathcal{G}/\mathcal{M}$-measurable function? I am looking forward to your answers :)


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