A question relating measure theory and Hilbert spaces

Question

Let $(w_n)$ be a sequence of positive real numbers and let $\ell^2(w)=\{(x_n):\sum\limits_{n=1}^{\infty}w_n|x_n|^2<\infty\}$. I have shown that this is a Hilbert space with the inner product defined by $\langle (x_n),(y_n)\rangle=\sum\limits_{n=1}^{\infty}w_nx_n\overline{y_n}$. Now there is question regarding $\ell^2(w)$ that I could not answer.

Is there a measure space $(X,\mathcal A,\mu)$ for which $\ell^2(w)$ is naturally identified with $L^2(\mu)$? Please help!

Answer 1

Hint: Let $X = \mathbb{N}_+ = \{1,2,3,\dotsc\}$, $\mathcal{A} = \mathcal{P}(X)$ and consider $$ \mu(A) = \sum_{n \in A} w_n \text{ for }A \in \mathcal{A}. $$