In Ergodic Theory, some (though not all) presentations of compact extensions use Hilbert bundles.

Given (X,B,μ,T) and a sub σ-algebra G⊆A one has an associated factor map π:(X,μ,T)→(Y,ν,T) (where π(x)=π(x′) iff (x∈A↔x′∈A) for every A∈G) and a decomposition μ=∫μydν (where ν=π∗(μ)).Then one can informally think of the space of functions in L2(X,μ) as an “integral” over the spaces of functions Hy:=L2(X,μy) for ν-a.e. y. Moreover, the union ∐yHy defines a Hilbert bundle over Y (of course, Y a priori has no topological structure, so this is not strictly precise).

This is the approach taken in these notes by Yuri Lima and in Ch. 9 of Glasner’s book, for instance.One then considers L∞(Y)-submodules (i.e. closed L∞(Y,ν)-invariant subspaces) of L2(X,μ). Now, here comes my problem. In proving many of the properties that one wants to prove for any such submodule M (I won’t go into these details or into compact extensions themselves) the authors usually consider the collection of {My}y∈Y, which are thought of as closed subspaces of the Hy defined above (i.e. {My}y∈Y is a subbundle of the Hilbert bundle). If f∈M, then fy is in My, where fy is just f “seen” as an element of L2(X,μy) instead of L2(X,μ) (warning: this is not the conditional expectation μy(f)). For instance, if we fix topological models such that X=Y×U then fy=f(y,⋅) The problem is that f is not a function, it is an equivalence class of functions. Certainly, for a fixed f∈L2(Y×U) the function fy=f(y,⋅) is defined for ν-a.e. y. But that doesn’t mean we can actually define a map M↦{My}ν-a.e.y for any closed subspace M, since such subspaces have uncountably many f‘s.

Question: Is there a way around this technical problem?

**Answer**

**Attribution***Source : Link , Question Author : Moritz , Answer Author : Community*