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Can someone help with the following? I have a continuous function $g: A_i \times A_{-i} \to \mathbb{R}^k$, and a probability measure $\mu \in \Delta(A_{-i})$. We can let $A_i=\mathbb{R}^n$ and $A_{-i}=\mathbb{R}^m$ to keep things simple, with the usual Borel Algebras. Is the following true ?\begin{equation} \int_{A_{-i}} g(a_i,a_{-i})d\mu(a_{-i}) \in g_{a_i}(A_{-i}) \end{equation} In other words, is there a vector $x$ s.t. $ g_{a_i}(x)= \mathbb{E}_{\mu_i} (g_{a_i}(a_{-i}))$?

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  • $\begingroup$ in general, you need the closed convex hull of $g_{a_i}(A_{-i})$ on the right hand side $\endgroup$ – user251257 Jul 29 '15 at 10:52
  • $\begingroup$ Thanks. Can you suggest any reference? $\endgroup$ – Enrico Jul 29 '15 at 11:11
  • $\begingroup$ There was a similar question recently. unfortunately I can't find it. Proof can be found in most functional analysis books. Basically, you assume the converse and separate the integral from the convex full with a linear functional (by Hahn Banach). If you push the linear functional into the integral, you obtain a contradiction. $\endgroup$ – user251257 Jul 29 '15 at 11:28
  • $\begingroup$ btw. not all continuous functions are integrable! $\endgroup$ – user251257 Jul 29 '15 at 11:28

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