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