Measurable structures for direct integrals

Question

I'm working with the notion of direct integrals as in Dixmier. Briefly: Given a measurable space $X$ and a family of separable Hilbert spaces $(H_x)_{x\in X}$, a measurable structure is a subspace $Y$ of the sections $\Pi_x H_x$ which satisfies the following axioms:

(1) for all $u\in Y$, $x\mapsto \|u(x)\|_{H_x}$ is measurable,

(2) given a section $u$, if for all $v\in Y$ one has that $x\mapsto (u(x),v(x))_{H_x}$ is measurable, then already $u\in Y$,

(3) there is a sequence $(u_n)_n$ in $Y$ such that for each $x\in X$, the sequence $(u_n(x))_n$ is dense in $H_x$.

If the family of spaces is constant, that is to say $H_x = H$ for some fixed separable Hilbert space $H$, then the measurable functions from $X$ to $H$ are a measurable structure.

Question: Are there other examples of measurable structures for such a constant family of spaces?

The maximality condition (2) makes it hard for me to come up with further examples. For instance, I tried to use the measurable functions in a closed subspace of $H$, but then condition (2) would force that all functions with values in the orthogonal complement were measurable as well.

Answer 1

Up to isomorphism, there are no other examples. But literally speaking, there are other examples. You could choose a non-measurable map $U$ from $X$ to the unitary group $\mathcal{U}(H)$, for instance by fixing a nontrivial unitary $U_0 \in \mathcal{U}(H)$ and putting $U(x) = U_0$ when $x$ belongs to a nonmeasurable set, while $U(x) = 1$ when $x$ belongs to the complement. Then you define $Y$ as the space of sections of the form $x \mapsto U(x) u(x)$, where $u$ is a measurable function from $X$ to $H$.