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 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.

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 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.

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.

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

Compactness in Lebesgue–Bochner spaces of random variables and the existence of mean-square random attractors

2019 ◽  
Vol 19 (04) ◽  
pp. 1950032
Author(s):  
Yejuan Wang ◽  
Xiangming Zhu ◽  
Peter Kloeden
Keyword(s):  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Separable Banach Space ◽  
Additional Condition ◽  
Probability Space ◽  
Parabolic Partial Differential Equations ◽  
Mean Square ◽  
Probabilistic Argument ◽  
Random Attractors

Let [Formula: see text] be a probability space and let [Formula: see text] be a separable Banach space. It is shown a subset [Formula: see text] of [Formula: see text], where [Formula: see text], is relatively compact in [Formula: see text] if and only if it is uniformly [Formula: see text]-integrable and uniformly tight. The additional condition of scalarly relatively compact required in the literature is shown to hold by a probabilistic argument. The result is then used to establish the existence of a mean-square random attractor for dissipative stochastic differential equations and stochastic parabolic partial differential equations.

scholarly journals Best Simultaneous approximation of quasi-continuous functions by monotone functions

1991 ◽  
Vol 50 (3) ◽  
pp. 391-408 ◽  
Author(s):  
Salem M. A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Continuous Functions ◽  
Unit Interval ◽  
Closed Convex Cone ◽  
Monotone Functions ◽  
Measurable Functions ◽  
Best Simultaneous Approximation ◽  
Nondecreasing Functions

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

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