A continuity in the weak topologies of a vector integral operator

Let $Y$ be a sequentially $\sigma(Y,X)$-complete Banach space satisfying the Radon-Nikodym property. Then, every vector integral operator $A:L_X\rightarrow M_U$ is continuous in the weak topologies $\sigma(L_X,L^\ast_Y )$ and $\sigma(M_U,M^\ast_V )$.