**The Question:** Is the following true? If not, what further hypotheses do I need?

> 
> Let $X$ be a Banach space, and let $C \subset X$ be closed and convex. Let $P$ be a probability measure over $D$, and let $f:D \to X$ with $f(D) \subseteq C$ be $P$-integrable. Then
>  $$
> \int\_D f \,dP \in C
> $$
> 
> 
> 

Intuitively, I believe this should be true since it is a sort of "limit" of convex combinations of elements of $C$. That is, when $P$ is a discrete measure with finitely many atoms, then this is simply the definition of convexity, and I would think that there is some limit by which we can reach integrals with respect to arbitrary measure.

I am not sure, however, how to formalize this argument. This seems like something that would pop up in some handy reference, or failing that, it at least it seems like a standard kind of argument of which someone here could fill in the blanks.

At the very least, I would be interested in an argument that works in the case that $X = \Bbb R^n$ (or $X = \Bbb C^n$).