scholarly journals Topological measure theory for completely regular spaces and their projective covers

1979 ◽  
Vol 82 (2) ◽  
pp. 565-584 ◽  
Author(s):  
Robert Wheeler
Keyword(s):  
Measure Theory ◽  
Completely Regular ◽  
Topological Measure ◽  
Topological Measure Theory ◽  
Projective Covers

Strict Topologies for Vector-Valued Functions

1974 ◽  
Vol 26 (4) ◽  
pp. 841-853 ◽  
Author(s):  
Robert A. Fontenot
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Measure Theory ◽  
Strict Topology ◽  
Locally Compact ◽  
Topological Measure ◽  
Topological Measure Theory ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Compact Hausdorff Space

This paper is motivated by work in two fields, the theory of strict topologies and topological measure theory. In [1], R. C. Buck began the study of the strict topology for the algebra C*(S) of continuous, bounded real-valued functions on a locally compact Hausdorff space S and showed that the topological vector space C*(S) with the strict topology has many of the same topological vector space properties as C0(S), the sup norm algebra of continuous realvalued functions vanishing at infinity. Buck showed that as a class, the algebras C*(S) for S locally compact and C*(X), for X compact, were very much alike. Many papers on the strict topology for C*(S), where S is locally compact, followed Buck's; e.g., see [2; 3].

The Strict Topology in a Completely Regular Setting: Relations to Topological Measure Theory

1972 ◽  
Vol 24 (5) ◽  
pp. 873-890 ◽  
Author(s):  
Steven E. Mosiman ◽  
Robert F. Wheeler
Keyword(s):  
Measure Theory ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Strict Topology ◽  
Completely Regular ◽  
Borel Measures ◽  
Topological Measure Theory ◽  
Locally Convex Topologies ◽  
Bounded Continuous Functions ◽  
Locally Convex

Let X be a locally compact Hausdorff space, and let C*(X) denote the space of real-valued bounded continuous functions on X. An interesting and important property of the strict topology β on C*(X) was proved by Buck [2]: the dual space of (C*(X), β) has a natural representation as the space of bounded regular Borel measures on X.Now suppose that X is completely regular (all topological spaces are assumed to be Hausdorff in this paper). Again it seems natural to seek locally convex topologies on the space C*(X) whose dual spaces are (via the integration pairing) significant classes of measures.

scholarly journals On lattice-perfect measures

1994 ◽  
Vol 56 (2) ◽  
pp. 169-182
Author(s):  
P. D. Stratigos
Keyword(s):  
Measure Theory ◽  
General Properties ◽  
Topological Measure ◽  
Topological Measure Theory ◽  
Countable Compactness ◽  
The Relationship ◽  
Lattice Measure

AbstractThe general properties of lattice-perfect measures are discussed. The relationship between countable compactness and measure perfectness, and the relationship between lattice-measure tightness and lattice-measure perfectness are investigated and several applications in topological measure theory are given.

Fubini and martingale theorems in C* -algebras

1979 ◽  
Vol 86 (1) ◽  
pp. 57-69
Author(s):  
C. J. K. Batty
Keyword(s):  
Measure Theory ◽  
Integration Theory ◽  
Affine Function ◽  
Radon Measures ◽  
C Algebras ◽  
Topological Measure Theory ◽  
Continuous Complex ◽  
Second Dual ◽  
Image Position ◽  
Complex Valued

The basic integration theory of Radon measures on locally compact spaces, as described in (5), has been developed in various directions both in commutative and non-commutative analysis. Thus if Ω is a compact Hausdorff space, and C(Ω) denotes the space of continuous complex-valued functions on Ω, Kaplan (16) showed how topological measure theory can be performed in the second dual C(Ω)** of C(Ω). Here as usual C(Ω)* is identified with the space of Radon measures on Ω by associating with a measure μ the linear functional φμ whereThus if f is a bounded real-valued function on Ω, there is an affine function f* defined on the set P(Ω) of Radon probability measures μ on Ω, byand f* extends by linearity to a functional in C(Ω)**. Conversely any function x on P(Ω) determines a function x0 on Ω. bywehere εω is the unit point of mass at ω.

Topological Extension Properties and Projective Covers

1982 ◽  
Vol 34 (6) ◽  
pp. 1255-1275 ◽  
Author(s):  
Haruto Ohta
Keyword(s):  
Topological Property ◽  
Extension Property ◽  
Discrete Space ◽  
Regular Space ◽  
Completely Regular ◽  
Projective Covers

Introduction. All spaces considered in this paper are assumed to be (Hausdorff) completely regular, and all maps are continuous. Let be a topological property of spaces. We shall identify with the class of spaces having . A space having is called a -space, and a subspace of a -space is called a -regular space. The class of -regular spaces is denoted by R(). Following [37], we call a closed hereditary, productive, topological property such that each -regular space has a -regular compactification a topological extension property, or simply, an extension property. In this paper, we restrict our attention to extension properties satisfying the following axioms:(A1) The two-point discrete space has .(A2) If each -regular space of nonmeasurable cardinal has , then = R().

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