A note on the Hall–Mergelyan theme

Let $\mu$ be a measure supported by the points $e^k$, $k=1,2,\dots,$ with the weights $\mu_k=e^{sk^2/2}$ where $s>1$ is a parameter. Then the polynomials are dense in the space$\mathcal L^p(\mu)$ for $p<s$ and are not dense in the space $\mathcal L^p(\mu)$ for $p<s$. This answers the question posed by Christian Berg and Jens Peter Reus Christensen.