Computing gaussian & exponential measures of semi-algebraic sets

We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $mu(om)$ of (not necessarily compact) basic semi-algebraic sets$omsubsetR^n$. We obtain two monotone (non increasing and non decreasing) sequences of upper and lower bounds $(overline{omega}_d)$, $(underline{omega}_d)$, $dinN$, each converging to $mu(om)$ as $dtoinfty$. For each $d$, computing $overline{omega}_d$ or $underline{omega}_d$reduces to solving a semidefinite program whose size increases with $d$. Some preliminary (small dimension) computational experiments are encouraging and illustrate thepotential of the method. The method also works for any measure whose moments are known and which satisfies Carlemans condition.