**tl;dr**: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?

I am essentially asking for the most general, infinite-dimensional analogue of this related question.

**Update:** I forgot to specicy that $K$ is compact. As @GeraldEdgar points out below, for noncompact $K$, the answer is trivially "no".

Suppose $K\subset E$ where $E$ is a topological vector space (as far as I can tell, this is the most general kind of space for which this question makes sense). Obviously we can define the closed convex hull $\overline{\text{conv} K}$ of $K$ as usual. Now consider the set $$ K^* = \{ \int_K x\,d\mu(x) : \mu \in\mathcal{P}(K)\}, $$ where $\mathcal{P}(K)$ is the set of (say, Borel) probability measures over $K$ and integral here is to be understood in the weak (Pettis) sense.

I would like to know when $\overline{\text{conv} K} = K^*$. If $E$ is finite-dimensional, there is equality. *What are the most general assumptions on $E$ and $K$ for which this equality continues to hold?*

(For the curious, the inspiration for this question came from trying to understand when $K^*$ is compact.)