A Riemann-type Integral that Includes Lebesgue-Stieltjes, Bochner and Stochastic Integrals

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American Mathematical Society, 1969 - Mathematics - 54 pages
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In this memoir we study the integrals defined as the limits of Riemann sums Summation(U(x sub i, A sub i)) in which the class of permitted pairs (x, A) and the nature of the limit process are subjected only to weak hypotheses. For the definition and the earlier theorems the assumptions are quite weak. To prove the deeper theorems, such as the monotone and dominated convergence theorems, we assume stronger hypotheses. However, they remain weak enough to allow us to obtain the Lebesgue-Stieltjes integral and the Bochner integral over locally compact domains, and also a generalization of K. Ito's stochastic integral, as well as several other integrals that have appeared in the literature. (Author).

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