# Measurable structures for direct integrals

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.

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 $$H$$.