Measurable (or continuous) orthonormal basis in a Hilbert space

Question

As you might know, the notion of measurable (or continuous) Parseval frame in a Hilbert space $H$ with respect to the measured space $(X,\Sigma,\mu)$ is defined to be a measurable function $f : (X,\Sigma) \to H$ such that for all $v \in H$, $\int_X | \langle v , f(x) \rangle |^2 d\mu(x) = ||v||^2$. An orthonormal basis $(f(x))_{x \in X}$ of $H$ indexed by $X$ is a continuous Parseval frame for the counting measure $\delta$. I would like to know if it could be a measurable Parseval frame for a different measure. Thanks.

Answer 1

Suppose your $f$ is a measurable Parseval frame for some measure $\mu$. Fix $x_0 \in X$. Then applying the definition with $v=f(x_0)$ we have $\langle f(x_0), f(x) \rangle = 1$ when $x=x_0$ and $0$ otherwise, so $$\mu(\{x_0\}) = \int_X |\langle f(x_0), f(x)\rangle|^2\,d\mu(x) = \|f(x_0)\|^2 = 1$$ and thus $\mu$ must be counting measure.