A continuity in the weak topologies of a vector integral operator (Q1366583)

Let \(\langle X,Y\rangle\) and \(\langle U,V\rangle\) be the pairs of Banach spaces in duality and \(Y\) be a sequentially \(\sigma(Y,X)\)-complete Banach space satisfying the Radon-Nikodym property. Let \(\langle\langle L_X,L_Y^*\rangle\rangle\) and \(\langle\langle M_U,M_V^*\rangle\rangle\) be the pairs of perfect spaces of measurable Banach-value functions in duality with respect to bilinear forms \[ \int_S\langle f(s),f^*(s)\rangle d\mu(s), \qquad \int_T\langle g(t),g^*(t)\rangle d\nu(t). \] In this case any vector integral operator \(A:L_X\to M_U\), defined by \[ (Af)(t)= \int_S K(s,t)f(s)d\mu(s), \quad \nu\text{-a.e.}, \quad f\in L_X, \] with a Bochner's measurable operator-valued kernel \(K:S\times T\to B_\sigma(X,U)\), satisfying the condition \[ \int_X\| K(s,t)\|\cdot\| f(s)\| d\mu(s)<\infty, \quad \nu\text{-a.e.}, \quad f\in L_X, \] is continuous in the weak topologies of \(\sigma(L_X,L_Y^*)\) and \(\sigma(M_U,M_V^*)\).