Let $f = 1_{[a,+\infty)}$ be the indicator function of a half-line. Does there exist a sequence $(P_n)$ of polynomials such that $f(x) \leq P_n(x)$ for every real $x$ and $$ \lim_{n\to \infty} \int_{\mathbf{R}} P_n d\gamma = \int_{\mathbf{R}} f d\gamma, $$ where $\gamma$ denotes the usual Gaussian measure ?