The Gaussian measure on algebraic varieties

We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety ${\displaystyle M\subset {ℝ}^n}$ is dense in the Hilbert space ${\displaystyle L^2(M,e^{-{\left|x\right|}^2}d\mu )}$, where dμ denotes the volume form of M and ${\displaystyle d\nu =e^{-{\left|x\right|}^2}d\mu }$ the Gaussian measure on M.