2
$\begingroup$

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.

$\endgroup$

1 Answer 1

4
$\begingroup$

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 $H$.

$\endgroup$
2
  • 1
    $\begingroup$ This is indeed a great example how to come up with other "different" measurable structures! Could you further explain your first assertion on uniqueness up to isomorphisms? $\endgroup$ Oct 23, 2021 at 15:24
  • 1
    $\begingroup$ Yes, by Lemma 8.12 and Theorem 8.13 in Takesaki's "Theory of operator algebras. Volume I.", two measurable fields of separable Hilbert spaces over a standard measure space are isomorphic if and only if the dimensions are equal almost everywhere. So the "other examples" that I suggested are the only examples with constant fibers. $\endgroup$ Oct 23, 2021 at 16:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.