Algebraic Versus Analytic Density of Polynomials

We show that under very mild conditions on a measure $\mu$ on the real line, the span of $\{x^n{\}}_{n=j}^{\mathrm{\infty }}$ is dense in $L^2(\mu )$ for any $j\in \mathbb{N}$. We also present a slightly weaker result with an interesting proof that uses Sobolev orthogonality.