Let $T = \smallint _Z^ \oplus T(\mathcal {E})$ be a direct integral of Hilbert space operators, and equip the collection $\mathcal {C}$ of compact subsets of C with the Hausdorff metric topology. Consider the [set-valued] function sp which associates with each $\mathcal {E} \in Z$ the spectrum of $T(\mathcal {E})$. The main theorem of this paper states that sp is measurable. The relationship between $\sigma (T)$ and $\{ \sigma (T(\mathcal {E}))\}$ is also examined, and the results applied to the hyperinvariant subspace problem. In particular, it is proved that if $\sigma (T(\mathcal {E}))$ consists entirely of point spectrum for each $\mathcal {E} \in Z$, then either T is a scalar multiple of the identity or T has a hyperinvariant subspace; this generalizes a theorem due to T. Hoover.