Why a random variable in a convex set has its mean in the set too

Question

Let $C$ be a convex set in $\mathbb{R^n}$, let $X$ be a random variable taking values in $C$, how do I show that its expectation, if it exists, is in $C$.

This claim appears in chapter 2 of the book convex optimization, but I didn't see a proof.

Answer 1

For a discrete random variable, the mean is simply a convex combination of the values. For a continuous r.v., see https://mathoverflow.net/questions/164836/is-an-integral-against-a-probability-measure-in-the-convex-hull-of-the-range