Expectation of a continuous function

Question

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}))$?