I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and Riemannian Manifolds) use the usual definition of a vector bundles to define Banach Bundles. As such, they usually do not need to worry about measurability issues. Due to presence of the locally trivializing map in their construction, given a connected base space, all the fibers are isomorphic to each other. Therefore, fibers with varying dimensions are not allowed. 


Other authors, like [Dautray and Lions](https://books.google.co.in/books?id=jlNtCQAAQBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false), or [Birman and Solomjak](https://books.google.co.in/books?id=unPrCAAAQBAJ&pg=PA297&dq=birman%20solomjak&hl=en&sa=X&ved=0ahUKEwiUmqORgKzMAhURBo4KHSvADMUQ6AEIKDAC#v=onepage&q=birman%20solomjak&f=false) define measurable Hilbert bundles and do not seem to insist on isomorphic fibers. 


My question is, in the case of isomorphic fibers, why not simply use Bochner spaces like $C(X;L^2(\Omega))$? Reed and Simon (in the fourth volume) insist that the focus is on the fibers rather than on $X$ and promise to cover general Hilbert bundles in Chapter XVI of their series "Methods of Modern Mathematical Physics", but as far as I know, that chapter never appeared.


Some other questions:


1. Is the "constant rank" condition necessary in order to put a differentiable (or continuous) structure on the bundle?
2. In contrast, is it true that the "constant rank" condition is not imposed on measurable Hilbert bundles because measurability is a weaker condition that allows for variation of fibers?
3. Are "constant rank" fibrations more "natural" in some way in mathematics?



Definitions of Hilbert Bundles
"Why not simply use Bochner spaces like $C(X;L^2(\Omega))$?" --- do you mean that this would be the space of continuous sections of the bundle with fiber $L^2(\Omega)$? Yes, that is correct if the bundle is *trivial*, i.e., a bundle of the form $X \times H$ where $H$ is the fiber Hilbert space. But of course not all bundles have this form. A natural class of examples where the bundle is usually not trivial arises when $X$ is a Riemannian manifold and the fiber at a point is the tangent space there. Each tangent space carries an inner product, so this is a Hilbert bundle.


"Is the constant rank condition necessary in order to put a continuous structure on the bundle?" --- yes. It's hard to see how to reasonably topologize a bundle whose fibers could vary in dimension. One does sometimes consider "dimension drop" conditions where you only consider sections which, say, vanish at some point, or lie in a proper subspace of the fiber at some point.


"Is it true that the constant rank condition is not imposed on measurable Hilbert bundles because measurability is a weaker condition that allows for variation of fibers?" --- again, I basically agree. This setting is very different from the topological setting; measurable sections will be totally insensitive to any global topological features of a bundle. So one can just take a measurable Hilbert bundle over a measure space $X$ to be something of the form $\bigcup (X\_n\times H\_n)$, with $n$ ranging over $\mathbb{N} \cup \{\infty\}$, $(X\_n)$ a measurable partition of $X$, and $H\_n$ an $n$-dimensional Hilbert space. (The nonseparable setting introduces some bad pathology, so I prefer to stick to the separable case.)


While I'm on the subject, measurable Hilbert bundles provide a setting for spectral theory that nicely accomodates multiplicity:


$\bullet$ If $A$ is a (bounded or unbounded) self-adjoint operator on a Hilbert space $H$, then there is a measurable Hilbert bundle over the spectrum of $A$, and an isometric isomorphism between $H$ and the $L^2$ sections of this bundle which turns $A$ into multiplication by $x$.


$\bullet$ If $\mathcal{A} \subset B(H)$ is a separable abelian C\*-algebra then there is a metrizable locally compact Hausdorff space $X$, a Borel measurable Hilbert bundle over $X$, and an isometric isomorphism between $H$ and the $L^2$ sections of the bundle which turns $\mathcal{A}$ into the set of multiplication operators by functions in $C\_0(X)$.


$\bullet$ If $\mathcal{M} \subseteq B(H)$ is an abelian von Neumann algebra, then there is a metrizable locally compact Hausdorff space $X$, a Borel measurable Hilbert bundle over $X$, and an isometric isomorphism between $H$ and the $L^2$ sections of the bundle which turns $\mathcal{M}$ into the set of multiplication operators by functions in $L^\infty(X)$.


I would say that these statements *cleanly exhibit the way the abstract C${}^\*$- and von Neumann algebras are situated within $B(H)$*.


(Details can be found in my book [Measure Theory and Functional Analysis](http://rads.stackoverflow.com/amzn/click/981450856X).)


