scholarly journals Completeness of the set of sub-σ-algebras

1993 ◽  
Vol 16 (3) ◽  
pp. 511-514
Author(s):  
Xikui Wang
Keyword(s):  
Functional Analysis ◽  
Metric Space ◽  
Probability Space ◽  
Point Of View ◽  
Complete Probability Space ◽  
Complete Probability ◽  
Analysis Point

A new metric is introduced on the set of all sub-σ-algebras of a complete probability space from functional analysis point of view. In this note, we will show that the resulting metric space is complete.

scholarly journals A metric space associated with probability space

1993 ◽  
Vol 16 (2) ◽  
pp. 277-282 ◽  
Author(s):  
Keith F. Taylor ◽  
Xikui Wang
Keyword(s):  
Metric Space ◽  
Probability Space ◽  
Von Neumann ◽  
Complete Probability Space ◽  
Complete Probability

For a complete probability space(Ω,∑,P), the set of all complete sub-σ-algebras of∑,S(∑), is given a natural metric and studied. The questions of whenS(∑)is compact or connected are awswered and the important subset consisting of all continuous sub-σ-algebras is shown to be closed. Connections with Christensen's metric on the von Neumann subalgebras of a TypeII1-factor are briefly discussed.

scholarly journals Convergence of Weak*-Scalarly Integrable Functions

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 112
Author(s):  
Noureddine Sabiri ◽  
Mohamed Guessous
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Separable Banach Space ◽  
Probability Space ◽  
Classical Theorem ◽  
Dual Vector ◽  
Integrable Functions ◽  
Complete Probability Space ◽  
Complete Probability ◽  
Bounded Sequences

Let (Ω,F,μ) be a complete probability space, E a separable Banach space and E′ the topological dual vector space of E. We present some compactness results in LE′1E, the Banach space of weak*-scalarly integrable E′-valued functions. As well we extend the classical theorem of Komlós to the bounded sequences in LE′1E.

scholarly journals A decomposition of bounded, weakly measurable functions

2011 ◽  
Vol 49 (1) ◽  
pp. 67-70
Author(s):  
Surjit Singh Khurana
Keyword(s):  
Banach Space ◽  
Measurable Function ◽  
Probability Space ◽  
Continuous Mapping ◽  
Measurable Functions ◽  
Separable Subspace ◽  
Complete Probability Space ◽  
Complete Probability

ABSTRACT Let (X,A,μ) be a complete probability space, ρ a lifting, Tρ the associated Hausdorff lifting topology on X and E a Banach space. Suppose F: (X,Tρ)-> E”σ be a bounded continuous mapping. It is proved that there is an A ∈ A such that FXA has range in a closed separable subspace of E (so FXA:X → E is strongly measurable) and for any B ∈ A with μ(B) > 0 and B ∩ A = ø, FXB cannot be weakly equivalent to a E-valued strongly measurable function. Some known results are obtained as corollaries.

Homomorphism-Compact Spaces

1983 ◽  
Vol 35 (3) ◽  
pp. 558-576 ◽  
Author(s):  
A. G. A. G. Babiker ◽  
S. Graf
Keyword(s):  
Probability Space ◽  
Topological Spaces ◽  
Measure Algebra ◽  
Point Mapping ◽  
Completely Regular ◽  
Compact Spaces ◽  
Strong Measure ◽  
Complete Probability Space ◽  
Complete Probability

In 1979 Edgar asked for a characterization of those completely regular Hausdorff topological spaces X which have the property that any Boolean σ-homomorphism from the Baire σ-field of X into the measure algebra of an arbitrary complete probability space can be realized by a measurable point-mapping. Those spaces X will be called homomorphism-compact or, for short, H-compact hereafter. It is wellknown that compact spaces are H-compact (cf. [4], p. 637, Proposition 3.4). We will show that the same is true for strongly measure compact spaces. On the other hand H-compact spaces are easily seen to be real-compact. Since the notions of measure-compactness and liftingcompactness (cf. [3]) also lie between strong measure-compactness and real-compactness it is natural to investigate the relations among these notions. Here the results are mainly negative (cf. Sections 4 and 6). Concerning the structural properties of H-compactness not very much can be said so far (cf. Section 7): it is, for instance, unknown whether the product of two H-compact spaces is again H-compact.

Evaluating norms of Pettis integrable functions

2009 ◽  
Vol 139 (6) ◽  
pp. 1255-1259
Author(s):  
M. Legua Fernández ◽  
L. M. Sánchez Ruiz
Keyword(s):  
Banach Space ◽  
Wide Class ◽  
Probability Space ◽  
Integrable Functions ◽  
Variation Norm ◽  
Complete Probability Space ◽  
Complete Probability

Assuming that (Ω, Σ, μ) is a complete probability space and that X is a Banach space, we evaluate both the semivariation and the variation norm of a wide class of Pettis integrable functions f : Ω → X.

scholarly journals HURWITZ THEOREM AND PARALLELIZABLE SPHERES FROM TENSOR ANALYSIS

2001 ◽  
Vol 16 (25) ◽  
pp. 4207-4222 ◽  
Author(s):  
J. A. NIETO ◽  
L. N. ALEJO-ARMENTA
Keyword(s):  
Clifford Algebra ◽  
Point Of View ◽  
Tensor Analysis ◽  
Tensor Algebra ◽  
Normed Algebra ◽  
Curved Spaces ◽  
Hurwitz Theorem ◽  
Analysis Point

By using tensor analysis, we find a connection between normed algebras and the parallelizability of the spheres S 1, S 3 and S 7. In this process, we discovered the analog of Hurwitz theorem for curved spaces and a geometrical unified formalism for the metric and the torsion. In order to achieve these goals we first develop a proof of Hurwitz theorem based on tensor analysis. It turns out that in contrast to the doubling procedure and Clifford algebra mechanism, our proof is entirely based on tensor algebra applied to the normed algebra condition. From the tersor analysis point of view our proof is straightforward and short. We also discuss a possible connection between our formalism and the Cayley–Dickson algebras and Hopf maps.

scholarly journals On statistical convergence in quasi-metric spaces

2019 ◽  
Vol 52 (1) ◽  
pp. 225-236 ◽  
Author(s):  
Merve İlkhan ◽  
Emrah Evren Kara
Keyword(s):  
Functional Analysis ◽  
Computer Science ◽  
Distance Function ◽  
Metric Space ◽  
Statistical Convergence ◽  
Triangle Inequality ◽  
Metric Spaces ◽  
Applied Mathematics ◽  
Comprehensive Investigation ◽  
Cauchy Sequences

AbstractA quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.

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