Hi,

It seems to be a common knowledge that the polynomials $x^n$ are dense in $L^2$ spaces with various probability weights, such as the gamma distribution weight $x^{\alpha-1}e^{-x}/\Gamma(\alpha)\;dx$.

Is there any reference to this fact preferrably including the condition which property of the weight implies the density of polynomials?

My guess is that if $$ \int_{-\infty}^{+\infty}e^{-\lambda|x|}P(dx)<\infty $$ for some $\lambda>0$, then the polynomials are dense in $L^2(\mathbb{R},P)$.