Suppose:
$\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $
$R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$.
$h : \mathbb{R}^d \to \mathbb{R}$ is compactly supported and continuous on its support such that $m( \partial \text{supp} \ h ) = 0$ where $m$ is $d$-dimensional Lebesgue measure.
$E = \{ x \in \mathbb{R}^d \ | \ c_1 \leq |x| \leq c_2 \}$ where $c_1,c_2 > 0$
Let $G: \mathbb{R}^d \to \mathbb{R}$ defined by
$G(S) = \mathbb{E}[ h( S \odot R) \ I_{E}(R) ] $
where $S \odot R := (S_1 R_1, ..., S_d R_d)$.
Does it follow that $m(\partial \ \text{supp} G) = 0$ ?
My thoughts: it is certainly true if $\text{supp} \ h$ is an interval in $\mathbb{R}$. One can compute the support in this case easily. I suppose for higher dimensional problems one will need to construct open covers of arbitrarily small measure, but this seems very difficult...
