On Integrable Functions in Complete Bornological Locally Convex Spaces

Abstract

A Lebesgue-type integration theory in complete bornological locally convex topological vector spaces was introduced by the first author in [17]. In this paper we continue developing this integration technique and formulate and prove some theorems on integrable functions as well as some convergence theorems. An example of Dobrakov integral in non-metrizable complete bornological locally convex spaces is given. (2010). Primary 46G10, Secondary 28B05. Mathematics Subject Classification

On Domination and Bornological Product Measures

“…[14]. (b) Δ u U,W the greatest δ-subring of Δ on which the restriction m U,W : Denote by Δ U,W ⊗ ∇ W,V the smallest δ-ring containing all rectangles A× B, A ∈ Δ U,W , B ∈ ∇ W,V , where (U, W ) ∈ U ×W, (W, V ) ∈ W ×V.…”

“…More on integrable functions and further results related to the generalized Dobrakov integral in C. B. L. C. S., see [18] and [19].…”

On Vector Integral Inequalities

“…[12]. Let (U, W ) ∈ U × W. We say that a charge m is of σ-finite (U, W )-semivariation if there exist sets E n ∈ ∆ U,W , n ∈ N, such that T = ∞ n=1 E n .…”

“…such that E n ∅ impliesm U,W (E n ) → 0. Further theorems on integrable functions and convergence theorems are proved in [15]. Proof.…”

“…Some theorems on integrable functions and convergence theorems for such an integral are proved in [15]. The construction and existence of product measures in C. B. L. C. S. in connection with this integration technique is given in [14].…”