Approximate Volume and Integration for Basic Semialgebraic Sets

Given a basic compact semialgebraic set $\mathbf{K}\subset\mathbb{R}^n$, we introduce a methodology that generates a sequence converging to the volume of $\mathbf{K}$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on $\mathbf{K}$ can be approximated as closely as desired, which permits the approximation of the integral on $\mathbf{K}$ of any given polynomial; the extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.