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VECTOR INTEGRALS AND PRODUCTS OF VECTOR MEASURES
Abstract
The main theme of this dissertation is to prove the existence of the product of two LCTVS-Valued (countably additive) measures in various cases, by expressing it as an "integral" (as in the classical case of non-negative measures). For this purpose, we first generalize Bartle's *-integral to LCTVS's. We take a triple (X,Y;Z) of LCTVS's together with a continuous bilinear mapping from X x Y into Z (called a bilinear system of LCTVS's), a measurable space (T, ) and a measure (beta): (--->) Y. For each continuous semi-norm r on Z and each bounded subset B of X, we define a semi-variation (VBAR)(VBAR)(beta)(VBAR)(VBAR)(,B,r) on by (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) the supremum being taken over all -partitions (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) of E and all (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The measure (beta) is said to have the "*-property" iff for each r, there exists a non-negative finite measure (nu)(,r) on such that, (VBAR)(VBAR)(beta)(VBAR)(VBAR)(,B,r) << (nu)(,r) for every B. If a single (nu) works (for all r), then (beta) has the "**-property". Finally (beta) has the "strong *-property" iff for each continuous semi-norm r on Z, there exist continuous semi-norms p on X and q on Y such that ((X,p),(Y,q);(Z,r)) is a bilinear system of semi-normed spaces and (beta) has *-property with respect to this system. Assuming that (beta) has *-property we construct an integral, integrating an X-valued function with respect to (beta); the integral lying in Z. We develop several properties of this integral culminating in a bounded convergence theorem when (beta) has **-property. We also construct a Bochner-type integral when the measure (beta) is of bounded variation. Using the above integration theories, we obtain the following results on products of LCTVS-valued measures. First a definition: An X-valued measure (alpha) is called "Mackey bounded" iff there exists a non-negative bounded set function (lamda) (on the domain of (alpha)) such that the set (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) is a bounded subset of X and for every E(,n) (DARR) (phi), (lamda)(E(,n)) (--->) 0. Theorem 1. If (alpha) is Mackey bounded and (beta) has **-property, then (alpha) x (beta) exists and (alpha) x (beta)(G) = (INT)(alpha)(G('t))d(beta)(t). Theorem 2. If (alpha) is Mackey bounded and (beta) has *-property, then (alpha) x (beta) exists. Theorem 3. If one of (alpha) and (beta) has strong *-property, then (alpha) x (beta) exists. Corollary. If one of (alpha) and (beta) is of bounded variation, then (alpha) x (beta) exists. Theorems 1 and 2 appear to be entirely new. The corollary generalizes Huneycutt (Studia Mathematica, XLI); who proved that (alpha) x (beta) exists when both (alpha) and (beta) are of bounded variation and the spaces are normed spaces. Theorem 3 generalizes and unifies the work of Duchon and Kluvanek (Mat. Cas. 17(1967)) where Z = X(' )(CRTIMES)(,(epsilon))(' )Y; the work of Duchon (Mat. Cas. 19(1969)) where Z = X(' )(CRTIMES)(,(pi))(' )Y; and of Swartz (Mat. Cas. 24 (1974)) where the bilinear map is assumed to be of "integral type". Finally we prove that (alpha) x (beta) inherits some properties possessed by both (alpha) and (beta).
Subject Area
Mathematics
Recommended Citation
SIVASANKARA, SASTRY A, "VECTOR INTEGRALS AND PRODUCTS OF VECTOR MEASURES" (1981). ETD collection for University of Nebraska-Lincoln. AAI8122603.
https://digitalcommons.unl.edu/dissertations/AAI8122603