Vrftjkry

up vote 1 down vote favorite Let $X$ be a compact metric space, and $Theta$ be a finite space, endowed with their own $sigma$ -algebra. Let $f colon X times Theta to mathbb{R}$ be a Caratheodory function such that (1) for each $x in X$ , the function $f(x, cdot) colon Theta to mathbb{R}$ is measurable; and (2) for each $theta in Theta$ , the function $f( cdot, theta) colon X to mathbb{R}$ is continuous. Given each $x in X$ , we have a probability distribution $pi( cdot ,| , x) colon 2^{Theta} to [0,1]$ . In particular, given any fixed $x in X$ , it will generate a corresponding probability distribution $pi$ on $2^Theta$ . I am curious that Under what kind of conditions (assumptions) imposed on this probability distribution $pi$ , the map $$X ni x mapsto int_Theta f(x,theta) , pi( mathrm{d} theta ,| ,x) in mathb...