Approximate volume and integration for basic semi-algebraic sets

Given a basic compact semi-algebraic set $\text{K}\subset {\mathbb{R}}^n$, we introduce a methodology that generates a sequence converging to the volume of $\text{K}$. 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 $\text{K}$ can be approximated as closely as desired, and so permits to approximate the integral on $\text{K}$ of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.