Fubini theorem w.r.t. stochastic measure on product measurable space

Using the new results about the existence of product S.M.^1], we get two forms of Fubini theorem about product S.M. on product measurable space in §1–§2. On being restricted to the special case of S.M. (I), the conditions needed are much weaker than those of 2] and couldn't be improved anymore. In the rest of this paper, we discuss how to calculate double integration w.r.t. non-product type S.M. on product space by iterated integration. Even in the case of classical measure theory, the problem hasn't been thoroughly solved yet.For the first two sections we suppose that (X, ), (Y, ), (, ) are measurable spaces, any two of which form a nice pair^1],P is a probability measure on , is theP-completion of , so (, ,P) is a complete probability space. LetL(, ,P) be the complete topological linear space which consists of all a.s. finite r.v. on (, ,P) (we identify those r.v. which differ only on a set of probability 0),L ₊(, ,P)={0, L(, ,P)}. IfZ, W areL ₊ (, ,P)-valued S.M. on , respectively, then there uniquely exists anL ₊(, ,P)-valued S.M. on , denoted byZ×W, such that

Keywords:	Product stochastic measure  non-product type S  M  enlarged type S  M
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