Semidefinite Relaxations for Lebesgue and Gaussian Measures of Unions of Basic Semialgebraic Sets

Given a finite Borel measure $\mu$ on ${\mathbb{R}}^n$ and basic semialgebraic sets ${\mathrm{\Omega }}_i\subset {\mathbb{R}}^n$, $i=1,\dots ,p$, we provide a systematic numerical scheme to approximate as closely as desired $\mu \left({\cup }_i{\mathrm{\Omega }}_i\right)$, when all moments of $\mu$ are available (and finite). More precisely, we provide a hierarchy of semidefinite programs whose associated sequence of optimal values is monotone and converges to the desired value from above. The same methodology applied to the complement ${\mathbb{R}}^n\setminus \left({\cup }_i{\mathrm{\Omega }}_i\right)$ provides a monotone sequence that converges to the desired value from below. When $\mu$ is the Lebesgue measure, we assume that $\mathrm{\Omega }≔{\cup }_i{\mathrm{\Omega }}_i$ is compact and contained in a known box $\mathbf{B}≔{\left[-a,a\right]}^n$, and in this case the complement is taken to be $\mathbf{B}\setminus \mathrm{\Omega }$. In fact, not only $\mu \left(\mathrm{\Omega }\right)$ but also every finite vector of moments of ${\mu }_{\mathrm{\Omega }}$ (the restriction of $\mu$ on $\mathrm{\Omega }$) can be approximated as closely as desired and so permits to approximate the integral on $\mathrm{\Omega }$ of any given polynomial.