  In this paper we investigate some reflexivity-type properties of separable
measurable Banach bundles over a $\sigma$-finite measure space. Our two main
results are the following:
  - The fibers of a bundle are uniformly convex (with a common modulus of
convexity) if and only if the space of its $L^p$-sections is uniformly convex
for every $p\in(1,\infty)$.
  - The fibers of a bundle are reflexive if and only if the space of its
$L^p$-sections is reflexive.
  These results generalise the well-known corresponding ones for
Lebesgue-Bochner spaces.
