  Let $X$ and $Y$ be Banach spaces and $(\Omega,\Sigma,\mu)$ a finite measure
space. In this note we introduce the space $L^p[\mu;L(X,Y)]$ consisting of all
(equivalence classes of) functions $\Phi:\Omega \mapsto L(X,Y)$ such that
$\omega \mapsto \Phi(\omega)x$ is strongly $\mu$-measurable for all $x\in X$
and $\omega \mapsto \Phi(\omega)f(\omega)$ belongs to $L^1(\mu;Y)$ for all
$f\in L^{p'}(\mu;X)$, $1/p+1/p'=1$. We show that functions in
$L^p[\mu;\L(X,Y)]$ define operator-valued measures with bounded $p$-variation
and use these spaces to obtain an isometric characterization of the space of
all $L(X,Y)$-valued multipliers acting boundedly from $L^p(\mu;X)$ into
$L^q(\mu;Y)$, $1\le q< p<\infty$.
