Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a convex compact subset of $\mathbb R^k$ (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 $\mathbb R^{d \times k}$. Consider the set $S \subseteq \mathbb R^d$ defined by

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

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 $\mathbb R^d$ ?