Representation of chance-constraints with strong asymptotic guarantees

Given $\epsilon \in (0,1)$, a probability measure $\mu$ on $\Omega\subset\mathbb{R}^p$ and a semi-algebraic set $K\subset X\times\Omega$, we consider the feasible set $X^*_\epsilon=\{x\in X:{\rm Prob}[(x,\omega)\in K]\geq 1-\epsilon\}$ associated with a chance-constraint. We provide a sequence of outer approximations $X^d_\epsilon=\{x\in X: h_d(x)\geq0\}$, $d\in\mathbb{N}$, where $h_d$ is a polynomial of degree $d$ whose vector of coefficients is an optimal solution of a semidefinite program. The size of the latter increases with the degree $d$. We also obtain the strong and highly desirable asymptotic guarantee that $\lambda(X^d_\epsilon\setminus X^*_\epsilon)\to0$ as $d$ increases, where $\lambda$ is the Lebesgue measure on $X$. Inner approximations with same guarantees are also obtained.