THE COMPOSITION OF OPERATOR-VALUED MEASURABLE FUNCTIONS IS MEASURABLE

Given separable Frechet spaces, E, F, and G, let L(E, F), L(F, G), and L(E, G) denote the space of continuous linear operators from E to F, F to G, and E to G, respectively. We topologize these spaces of operat ors by any one of a family of topologies including the topology of poi nt-wise convergence and the topology of compact convergence. We will s how that if (X, F) is any measurable space and both A:X --> L(E, F) an d B:X --> L(F, G) are Borelian, then the operator composition BA:X --> L(E, G) is also Borelian. Further, we will give several consequences of this result.