Let $X$ be a Banach space, and for any compactly supported Borel probability measure $\mathbb{P}$ on $X$, define the mean $\mu_\mathbb{P}$ by $\mu_\mathbb{P}=\int_X x \, \mathbb{P}(dx)$.

I want to say that a relatively compact Borel set $S \subset X$ is measure-theoretically convex if every Borel probability measure $\mathbb{P}$ with $\mathbb{P}(S)=1$ has $\mu_\mathbb{P} \in S$.

Is there a recognised term for what I have called "measure-theoretically convex"?

Note that

• if $S$ is measure-theoretically convex then $S$ is convex;
• if $S$ is compact and convex then $S$ is measure-theoretically convex.

To give some further examples, suppose that $X$ is the completion of the vector space of finite signed measures on a compact metric space $M$ equipped with a norm $\|\cdot\|$ whose topology is the topology of weak convergence. (E.g. one can find a sequence of continuous functions $g_n \colon M \to [0,2^{-n}]$ such that $\|\mu\|:=\sum_n |\int_M g_n \, d\mu|$ is such a norm.) Then,

• the set of all probability measures supported on a finite subset of $M$ is convex, but not measure-theoretically convex;
• given a non-closed Borel set $A \subset M$ and a non-empty interval $I \subset (0,1]$, the set of all probability measures assigning to $A$ a value in $I$ is not closed in $X$, but nonetheless is measure-theoretically convex.