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 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.

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 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.

Best L∞-approximation of measurable, vector-valued functions

1984 ◽  
Vol 96 (3) ◽  
pp. 477-481 ◽  
Author(s):  
Abdallah M. Al-Rashed ◽  
Richard B. Darst
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Convex Banach Space ◽  
Equivalence Classes ◽  
Uniformly Convex ◽  
Integrable Functions ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Measurable Vector

Let (Ω, ,μ) be a probability space, and let be a sub-sigma-algebra of . Let X be a uniformly convex Banach space. Let A =L∞(Ω, , μ X) denote the Banach space of (equivalence classes of) essentially bounded μ-Bochner integrable functions g: Ω.→ X, normed by the function ∥.∥∞ defined for g ∈ A by(cf. [6] for a discussion of this space). Let B = L∞(Ω, , μ X), and let f ε A. A sufficient condition for g ε B to be a best L∞-approximation to f by elements of B is established herein.

scholarly journals Spaces of vector functions that are integrable with respect to vector measures

2007 ◽  
Vol 82 (1) ◽  
pp. 85-109 ◽  
Author(s):  
José Rodríguez
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Equivalence Classes ◽  
The Other ◽  
Normed Spaces ◽  
Vector Measures ◽  
Other Hand ◽  
Integrable Functions ◽  
Compact Range ◽  
Indefinite Integral

AbstractWe study the normed spaces of (equivalence classes of) Banach space-valued functions that are Dobrakov,S* or McShane integrable with respect to a Banach space-valued measure, where the norm is the natural one given by the total semivariation of the indefinite integral. We show that simple functions are dense in these spaces. As a consequence we characterize when the corresponding indefinite integrals have norm relatively compact range. On the other hand, we also determine when these spaces are ultrabornological. Our results apply to conclude, for instance, that the spaces of Birkhoff (respectively McShane) integrable functions defined on a complete (respectively quasi-Radon) probability space, endowed with the Pettis norm, are ultrabornological.

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.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

scholarly journals On Simultaneous Farthest Points in

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Sh. Al-Sharif ◽  
M. Rawashdeh
Keyword(s):  
Banach Space ◽  
Bounded Subset ◽  
Farthest Points ◽  
Integrable Functions

Let be a Banach space and let be a closed bounded subset of . For , we set  . The set is called simultaneously remotal if, for any , there exists such that  . In this paper, we show that if is separable simultaneously remotal in , then the set of -Bochner integrable functions, , is simultaneously remotal in . Some other results are presented.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

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