scholarly journals Extensions of tight set functions with applications in topological measure theory

1984 ◽  
Vol 283 (1) ◽  
pp. 353-353 ◽  
Author(s):  
Wolfgang Adamski
Keyword(s):  
Measure Theory ◽  
Tight Set ◽  
Set Functions ◽  
Topological Measure ◽  
Topological Measure Theory

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].

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.

A SUPER RADON-NIKODYM DERIVATIVE FOR ALMOST SUBADDITIVE SET FUNCTIONS

2013 ◽  
Vol 21 (03) ◽  
pp. 347-365 ◽  
Author(s):  
YANN RÉBILLÉ
Keyword(s):  
Density Function ◽  
Measure Theory ◽  
Type Theorem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Set Functions ◽  
Necessary And Sufficient ◽  
Exact Factorization ◽  
Nikodym Derivative ◽  
Nikodym Theorem

In classical measure theory, the Radon-Nikodym theorem states in a concise condition, namely domination, how a measure can be factorized by another (bounded) measure through a density function. Several approaches have been undertaken to see under which conditions an exact factorization can be obtained with set functions that are not σ-additive (for instance finitely additive set functions or submeasures). We provide a Radon-Nikodym type theorem with respect to a measure for almost subadditive set functions with bounded disjoint variation. The necessary and sufficient condition to guarantee a superior Radon-Nikodym derivative remains the standard domination condition for measures. We show how these set functions admit an equivalent factorization under the standard domination condition for set functions.

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.

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.

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