Let $X$ be a compact space and let $A$ be a convex compact subset of $\mathbb R^d$ (e.g the probability simplex in $\mathbb R^d$). Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^d$. Consider the set $S \subseteq \mathbb R^d$ defined by

$$
S := \left\{\int_{X}\pi v\,\mathrm{d}P \mid \pi \in \Pi\right\},
$$

where $\Pi$ is the set of $P$-measurable functions from $X$ to $A$.


>**Question 1.** *Under what general conditions is $S$ a closed subset of $\mathbb R^d$ ?*

Perhaps even more generally,

>**Question 2.** *What is closure $\overline S$ of $S$ in $\mathbb R^d$ ?*