Bounded Measures on Topological Spaces

Section: Definitions and Preliminaries A Boolean Algebra And Is A Commmentioning

confidence: 99%

mentioning

confidence: 99%

An 𝐿¹-space for Boolean algebras and semireflexivity of spaces 𝐿^{∞}(𝑋,Σ,𝜇)

Sentilles¹

1977

“…Topoplogies ß0, ß, and ßx on Q(F) which yield A/,, Mr, and Ma, respectively, as dual spaces are introduced in [24], [58]. It is shown in [74] that ße = t(C6, Ms) is the finest locally convex topology on Cfc( T) which agrees with 9^ on every member of S.…”

Section: Basic Definitions and Terminology Throughout F Denotes A Cmentioning

confidence: 99%

“…Thus g is a oiCb, A/T)-cluster point of (/"), so H is relatively o(Cb, Accountably compact. Since (Cb(T), ß) is a complete LCS for metrizable F [24,Theorem 7] …”

Section: Basic Definitions and Terminology Throughout F Denotes A Cmentioning

confidence: 99%

Weak and pointwise compactness in the space of bounded continuous functions

Wheeler¹

1981

Abstract.Let T be a completely regular Hausdorff space, Cb(T) the space of bounded continuous real-valued functions on T, M(T) the Banach space dual of Cb(T). Let % denote the family of subsets of Cb(T) which are uniformly bounded and relatively compact for the topology 9^ of pointwise convergence. A well-known theorem of Grothendieck [28] asserts that if F is a compact Hausdorff space, and H is a uniformly bounded, pointwise compact set of real-valued continuous functions on F, then H is compact in the weak topology of the Banach space CÍT). Thus, in this case, the weak and pointwise topologies agree on//. This article is concerned with extensions of Grothendieck's Theorem to the setting: F is a completely regular Hausdorff space, and CbiT) is the Banach space of bounded continuous real-valued functions on T, with the supremum norm. Ptak [51] and Tomasek [64] have shown that Grothendieck's Theorem as stated above holds if and only if T is pseudocompact. However, one can pursue the matter as follows:Main Problem. For a given completely regular T, what is the largest linear subspace Z of MÎT), the Banach space dual of CbiT), such that the pointwise topology and the weak topology

“…From these beginnings, there has developed in recent years a rather substantial topological measure theory through the efforts of many investigators. Among them should be mentioned Fremlin, Garling and Haydon [4], E. Granirer [6], J. D. Knowles [10], W. Moran [11], [12], F. D. Sentüles [15], [16], [17], R. F. Wheeler [17], [20] and S. Mosiman [13]. Almost all of this work has been done in the context of a completely regular Hausdorff space, and the studies have been concerned with the algebra Cb of bounded, continuous real-valued functions and its dual space.…”

mentioning

confidence: 99%

A generalized topological measure theory

Kirk¹,

Crenshaw²

1975

ABSTRACT. The theory of measures in a topological space, as developed by V. S. Varadarajan for the algebra C of bounded continuous functions on a completely regular topological space, is extended to the context of an arbitrary uniformly closed algebra A of bounded real-valued functions. Necessary and sufficient conditions are given for A * to be represented in the natural way by a space of regular finitely-additive set functions. The concepts of additivity and tightness for these set functions are considered and some remarks about weak convergence are made.