Bochner Integration of Operator-Valued Functions in Terms of Discrete-Time Normal Martingales

Let M be a discrete-time normal martingale satisfying some mild conditions, \mathscrS(M)\subset L^2(M)\subset \mathscrS^*(M) be the Gel'fand triple constructed from the functionals of M. \mathscrL denote the space of continuous linear operators from the testing functional space \mathscrS(M) to the generalized functional space \mathscrS^*(M). As is known, the usual product in \mathscrL may not make sense. However, by using the 2D-Fock transform, one can introduce convolution in \mathscrL, then one can try to introduce a Bochner-style integral for \mathscrL-valued functions with respect to \mathscrL-valued measures in the sense of convolution. This paper just studies such a type of integral. First, a class of \mathscrL-valued measures are introduced and their basic properties are examined. Then, an integral of an \mathscrL-valued function with respect to an \mathscrL-valued measure is defined and a dominated convergence theorem is established for this integral. Finally, a convolution measure of two \mathscrL-valued measures is also discussed and a Fubini type theorem is proved for this integral.