VECTOR INTEGRALS AND PRODUCTS OF VECTOR MEASURES

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.;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).;(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).