$D = \{d_i: X\to Y, i=1,\dots,n\}$ is a finite set of mappings from $X$ to $Y$, $(\Omega, \mathcal F, P)$ is a probability space, and $\delta: \Omega \to D$ is a measurable mapping. Can $\delta$ induce some random measure on $Y$?

E.g. in decision theory, think $X$ as a sample space with $(\Omega, \mathcal F, P)$ as the underlying probability space, $Y$ as an action sapce, $D$ as a finite set of nonrandomized rules, and $\delta$ as a randomized rule. My question above comes from that I saw somewhere a randomized rule is defined as a random measure on the action space, but I couldn't connect this definition with the original definition "a measurable mapping from $\Omega$ to $D$".



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