# Expectation of a continuous function

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 ?$$\int_{A_{-i}} g(a_i,a_{-i})d\mu(a_{-i}) \in g_{a_i}(A_{-i})$$ In other words, is there a vector $x$ s.t. $g_{a_i}(x)= \mathbb{E}_{\mu_i} (g_{a_i}(a_{-i}))$?

• in general, you need the closed convex hull of $g_{a_i}(A_{-i})$ on the right hand side – user251257 Jul 29 '15 at 10:52
• Thanks. Can you suggest any reference? – Enrico Jul 29 '15 at 11:11
• 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. – user251257 Jul 29 '15 at 11:28
• btw. not all continuous functions are integrable! – user251257 Jul 29 '15 at 11:28