A general bilinear vector integral

Strongly Bounded Representing Measures and Convergence Theorems

“…Although the convergence theorems in [4] are similar to some of the conclusions in our first theorem, it is not clear that [4] can be used to obtain the specific results we desire. For this reason, as well as for the convenience of the reader, we include a brief description of the bilinear integral we shall use and a proof of the convergence results we need.…”

“…The general bilinear integral of Bartle [4] can be used in the context of strongly bounded representing measures to establish convergence results which unify several approaches and have numerous applications and corollaries. Although the convergence theorems in [4] are similar to some of the conclusions in our first theorem, it is not clear that [4] can be used to obtain the specific results we desire.…”

Linear Operators and Vector Measures

“…Now since m is s-bounded, K = {\m,\: z G fi*} is conditionally weakly compact in ca(2, C); thus by [13,IV.9.1] there is a positive A G ca(2, C) so that K < X uniformly and X(A) < s\ip{\mz\(A): z G F*). Therefore X(A) -> 0 if and only if m(A) -* 0, and À is a control measure for m in the sense of Bartle [1]. By the bilinear dominated convergence theorem in [1] it follows that Sfodm = limB/Sfli/«J = limB2,l.L(/).…”

“…Therefore X(A) -> 0 if and only if m(A) -* 0, and À is a control measure for m in the sense of Bartle [1]. By the bilinear dominated convergence theorem in [1] it follows that Sfodm = limB/Sfli/«J = limB2,l.L(/).…”

Representation theorems on Banach function spaces

“…p '. In order to do this, Bartle's treatment [2] of integration will be used. The characterization of iV* is in terms of certain additive set functions.…”