This question comes up in studying mean parametrization of exponential families of distributions. (See Brown's 1986 book on the subject.)

Let $\nu$ be a (Borel) measure on $\mathbb R^d$. Let $p(\cdot)$ denote a generic probability density w.r.t $\nu$, and let \begin{align} \mathcal M &:= \Big\{ \int x p(x) d\nu(x):\; \int p(x) d\nu(x) = 1\Big\} \\&:= \Big\{ \int x d P:\; P(\mathbb R^d) = 1, \quad P \ll \nu\Big\} \end{align} be the set of realizable means by probability measures absolutely continuous w.r.t. $\nu$.

Now, let $\text{supp}(\nu)$ be the support of $\nu$, the smallest closed set whose complement has $\nu$-measure zero. Let $$ \mathcal K = \overline{\text{conv}}(\text{supp}(\nu)) $$ where $\overline{\text{conv}}$ denotes the closed convex hull of a set. Brown calls $\mathcal K$ the convex support of $\nu$. (Is this a standard terminology?)

It seems that $\mathcal M$ and $\mathcal K$ are close, say $\mathcal M \subset \mathcal K \subset \overline{\mathcal M}$ (?) and the two could only be different over the boundaries of the two set. Any references shedding light on the relationship, esp. what happens at the boundary of these sets, is appreciated. (There seems to be a vague connection to Choquet theory?)

An example: Let $\nu$ be the push-forward of the 1-D Lebesgue measure by the map $x \mapsto (x,x^2)$, then the set $\mathcal M = \{(\mu_1,\mu_2) :\; \mu_2 > \mu_1^2 \}$ which is an open set. To see this, we note that $(\mu_1,\mu_2) \in \mathcal M$ iff $(\mu_1,\mu_2) = E (X,X^2)$ where $X$ is a random variable whose distribution is absolutely continuous w.r.t. the Lebesgue measure. We always have $\mu_2 \ge \mu_1^2$ (by Jensen inequality) and anything with $\mu_2 > \mu_1^2$ can be realized by a non-degenerate Gaussian distribution with mean $\mu_1$ and variance $\mu_2 - \mu_1^2$. Anything on the boundary $\mu_2 = \mu_1^2$ corresponds to a point mass (variance = 0), hence cannot be realized by a distribution absolutely continuous w.r.t. the Lebesgue measure.

In fact, the interesting result is that anything in interior of $\mathcal K$ (or shall we say interior of $\mathcal M$) can be realized by the corresponding exponential family. This is Theorem 3.6 in Brown. The above example is a special case of this, where the Gaussian is the corresponding exponential family distribution.