Integral representation of linear operators on Orlicz-Bochner spaces

Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measurement space and let $\mathcal{L} (X, Y)$ stand for the space of all bounded linear operators between Banach spaces $ (X, \Vert\cdot\Vert_X)$ and $ (Y, \Vert\cdot\Vert_Y)$ We study the problem of integral representation of linear operators from an Orlicz-Bochner space $L^\varphi (\mu, X)$ to $Y$ with respect to operator measures $m:\Sigma\longrightarrow \mathcal{L} (X, Y)$. It is shown that a linear operator $T:L^\varphi (\mu, X)\longrightarrow Y$ has the integral representation $T(f)=\int_\Omega f(\omega)dm$ with respectto a $\varphi*$ -variationally $\mu$-continuous operator measure $m$ if and only if $T$ is $(\gamma_\varphi,\Vert\cdot\Vert _Y)$-continuous, where $\gamma_\varphi$ stands for a natural mixed topology on $L^\varphi(\mu, X)$. As an application, we derive Vitali-Hahn-Saks type theorems for families of operator measures.