We show that under very mild conditions on a measure \(\mu\) on the interval \([0,\infty)\) , the span of \(\{x^k\}_{k=n}^{\infty}\) is dense in \(L^2(\mu)\) for any \(n=0,1,\ldots{}\) . We present two different proofs of this result, one based on the density index of Berg and Thill and one based on the Hilbert space \(L^2(\mu)\oplus \mathbb{C}^{n+1}\) . Using the index of determinacy of Berg and Durán we prove that if the measure \(\mu\) on \(\mathbb{R}\) has infinite index of determinacy then the polynomial ideal \(R(x)\mathbb{C}[x]\) is dense in \(L^2(\mu)\) for any polynomial R with zeros having no mass under \(\mu\) .
