Lebesgue and gaussian measure of unions of basic semi-algebraic sets

Abstract

Given a finite Borel measure $\mu$ on R n and basic semi-algebraic sets $\Omega$\_i $\subset$ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired $\mu$(\cup\_i $\Omega$\_i), 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 R n \ (\cup\_i $\Omega$\_i) provides a monotone sequence that converges to the desired value from below. When $\mu$ is the Lebesgue measure we assume that $\Omega$ := \cup\_i $\Omega$\_i is compact and contained in a known box B and in this case the complement is taken to be B \ $\Omega$. In fact, not only $\mu$($\Omega$) but also every finite vector of moments of $\mu$\_$\Omega$ (the restriction of $\mu$ on $\Omega$) can be approximated as closely as desired, and so permits to approximate the integral on $\Omega$ of any given polynomial.