subspace of $L^2$

Question

Let $(X,B(X),\mu)$ a measurable space, for a positive finite measure $\mu$, we consider $H=L^2(X,d\mu)$, Let $A$ a closed subspace of $H$, we know that $A$ is a hilbert space, can we say that it exist a positive measure $\mu'$ such that $A=L^2(X,d\mu')$ ? For example can we say that Hardy space $\mathbb{H}^2(\mathbb{T})$ is a $L^2$ space

Answer 1

If $B$ is an orthonormal basis for the Hilbert space $A$ then $A$ is isomorphic to $L^2(\#)$, where $\#$ is counting measure on $B$. (Commonly known as $\ell^2(B)$.) So yes, every Hilbert space is an $L^2$ space.

Can we make it $L^2(\nu)$ where $\nu$ is some measure on $X$? Yes, but not in any interesting way - $X$ has a subset with the same cardinality as $B$. This is really irrelevant; the measure doesn't really have anything to do with $X$ or $\mu$.