Closed convex hull in infinite dimensions vs. continuous convex combinations 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.)
 A: No.  Even in one dimension.  Say $K$ is the open interval $(0,1)$.  Show $0 \notin K^*$.  Let $\mu$ be a probability measure with support contained in $(0,1)$.  Indeed,
$$
r(\mu) := \int_K x\,d\mu(x)
$$
is the integral of a positive function.  That is, $x > 0$ a.e.  So $\int_K x\,d\mu(x) > 0$.  Similarly $1 \notin K^*$.
In a locally convex topological vector space  $E$, if there is any extreme point of $M = \overline{\text{conv} K}$ that does not already belong to $K$, then it also does not belong to $K^*$.  So what if $K$ is the set $\text{ex}\; M$ of extreme points of a closed convex bounded set $M$?  Can we recover $M$ as $K^*$?
A very nice little book that discusses this situation is
Phelps, Robert R., Lectures on Choquet’s theorem, Lecture Notes in Mathematics. 1757. Berlin: Springer. 124 p. (2001). ZBL0997.46005.
Choquet's theorem tells us roughly that every point of a compact convex set $M$ is of the form $r(\mu)$ for some probability measure concentrated on the set $\text{ex}\; M$ of extreme points of $M$.

Plug
My first publication to attract any notice was this one, where there is a generalization of Choquet's theorem to certain closed bounded noncompact sets $M$.
Edgar, G. A., A noncompact Choquet theorem, Proc. Am. Math. Soc. 49, 354-358 (1975). ZBL0273.46012.
