The Fubini theorems of stochastic measures (Q1113527)

Suppose that (S,\(\Sigma)\) is a measurable space, E a Banach space, and Z a vector measure on \(\Sigma\) with values in the dual E' of E. If \(f: S\to E\) is a simple function of the form \(f=\sum^{n}_{i=1}x_ i 1_{A_ i}\) \((x_ i\in E\), \(A_ i\in \Sigma\) disjoint), it is natural to define the integral of f relative to Z by \[ \int f dZ:=\sum^{n}_{i=1}<x_ i,Z(A_ i)>. \] Under suitable assumptions on Z (e.g. to be of bounded variation) it is possible to extend this intgral and to prove analogues to theorems of classical integration theory: dominated convergence theorem, Fubini theorem etc. The paper contains some results of this type for the case \(E=L^ p\). The proofs are straight-forward generalizations of the classical proofs.