The Bartle-dunford-schwartz Integral: Integration With Respect To A Sigma-additive Vector Measure (monografie Matematyczne)

This volume is a thorough and comprehensive treatise on vector measures. The functions to be integrated can be either [0,infinity]- or real- or complex-valued and the vector measure can take its values in arbitrary locally convex Hausdorff spaces. Moreover, the domain of the vector measure does not have to be a sigma-algebra: it can also be a delta-ring. The book contains not only a large amount of new material but also corrects various errors in well-known results available in the literature. It will appeal to a wide audience of mathematical analysts. In 1953, Grothendieck [G] characterized locally convex Hausdor? spaces which have the Dunford-Pettis property and used this property to characterize weakly compact operators u : C(K)? F,where K is a compact Hausdor? space and F is a locally convex Hausdor? space (brie?y, lcHs) which is complete. Among other results, he also showedthat there is a bijective correspondencebetween the family of all F-valued weakly compact operators u on C(K) and that of all F-valued ?-additive Baire measures on K. But he did not develop any theory of integration to represent these operators. Later, in 1955, Bartle, Dunford, and Schwartz [BDS] developed a theory of integration for scalar functions with respect to a ?-additive Banach-space-valued vector measure m de?ned on a ?-algebra of sets and used it to give an integral representationfor weakly compact operatorsu : C(S)? X,where S is a compact Hausdor? space and X is a Banach space. A modi?ed form of this theory is given inSection10ofChapterIVof[DS1].Inhonoroftheseauthors,we callthe integral introduced by them as well as its variants given in Section 2.2 of Chapter 2 and in Section 4.2 of Chapter 4, the Bartle-Dunford-Schwartz integral or brie?y, the BDS-integral. This volume is a thorough and comprehensive treatise on vector measures. The functions to be integrated can be either 0, infinity]- or real- or complex-valued and the vector measure can take its values in arbitrary locally convex Hausdorff spaces. Moreover, the domain of the vector measure does not have to be a sigma-algebra: it can also be a delta-ring. The book contains not only a large amount of new material but also corrects various errors in well-known results available in the literature. It will appeal to a wide audience of mathematical analysts