  In this paper we study derivations in subalgebras of $L_{0}^{wo}(\nu ;%
\mathcal{L}(X)) $, the algebra of all weak operator measurable funtions $f:S\to
\mathcal{L}(X) $, where $% \mathcal{L}(X) $ is the Banach algebra of all
bounded linear operators on a Banach space $X$. It is shown, in particular,
that all derivations on $L_{0}^{wo}(\nu ;\mathcal{L}(X)) $ are inner whenever
$X$ is separable and infinite dimensional. This contrasts strongly with the
fact that $L_{0}^{wo}(\nu ;\mathcal{L}(X)) $ admits non-trivial non-inner
derivations whenever $X$ is finite dimensional and the measure $\nu $ is
non-atomic. As an application of our approach, we study derivations in various
algebras of measurable operators affiliated with von Neumann algebras.
