Let \(\mu\) be a positive finite Borel measure defined on a \(\sigma\)-algebra of subsets of a set \(\mathbb X\). Let \(\phi =(\phi_0,\phi_1\dots,\phi_n):\mathbb X\to\mathbb R^{n+1}\) be a measurable function such that \(\phi_j^2\leq c(1+\phi_0^2)\) for \(j=1,\dots,n\) and some \(c>0\). Let \(p\) be a real non-constant polynomial such that the function \(1/(p\circ\phi_0)\) is bounded \(\mu\)-almost everywhere and let the algebra \[ D=\text{span} \{ \phi_0^{\alpha_0} \phi_1^{\alpha_1}\cdots\;\phi_n^{\alpha_n} (p\circ\phi_0)^{-\alpha_{n+1}} : \alpha_j\in\mathbb{N},\;j=0,1,\dots,n+1 \} \] be contained in \(L^2(\mu)\). It is proved that under these assumptions, the algebra \(D\) is dense in \(L^2(\rho\mu)\) for every nonnegative \(\rho\in L^2(\mu)\). Some applications are considered.