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.

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.

Exponential convergence rates for weighted sums in noncommutative probability space

2016 ◽  
Vol 19 (04) ◽  
pp. 1650027 ◽  
Author(s):  
Byoung Jin Choi ◽  
Un Cig Ji
Keyword(s):  
Probability Space ◽  
Von Neumann Algebra ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Large Deviation ◽  
Type Inequality ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Noncommutative Probability ◽  
Von Neumann

We study exponential convergence rates for weighted sums of successive independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. Then we prove a noncommutative extension of the result for the exponential convergence rate by Baum, Katz and Read. As applications, we first study a large deviation type inequality for weighted sums in a noncommutative probability space, and secondly we study exponential convergence rates for weighted free additive convolution sums of probability measures.

Weak convergence of sequences of random elements with random indices

1980 ◽  
Vol 88 (1) ◽  
pp. 171-174 ◽  
Author(s):  
M. Csörgő ◽  
Z. Rychlik
Keyword(s):  
Positive Integer ◽  
Weak Convergence ◽  
Metric Space ◽  
Probability Space ◽  
Random Element ◽  
Random Variables ◽  
Random Elements ◽  
Convergence Of Sequences ◽  
Random Indices ◽  
Separable Metric Space

Let (S, d) be a separable metric space equipped with its Borel σ field . Let {Yn, n ≥ 1} be a sequence of S-valued random elements defined on a probability space (Ω, , p). Assume Yn ⇒ Y converges weakly to an S-valued random element Y. Let {Nn, n ≥ 1} be a sequence of positive integer-valued random variables defined on the same probability space (Ω, , p).

scholarly journals Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

2022 ◽  
Vol 77 (1) ◽  
Author(s):  
Karol Baron ◽  
Rafał Kapica
Keyword(s):  
Metric Space ◽  
Probability Space ◽  
Law Of Large Numbers ◽  
Weak Limit ◽  
Compact Metric Space ◽  
Strong Law ◽  
Large Numbers

AbstractAssume $$ (\Omega , {\mathscr {A}}, P) $$ ( Ω , A , P ) is a probability space, X is a compact metric space with the $$ \sigma $$ σ -algebra $$ {\mathscr {B}} $$ B of all its Borel subsets and $$ f: X \times \Omega \rightarrow X $$ f : X × Ω → X is $$ {\mathscr {B}} \otimes {\mathscr {A}} $$ B ⊗ A -measurable and contractive in mean. We consider the sequence of iterates of f defined on $$ X \times \Omega ^{{\mathbb {N}}}$$ X × Ω N by $$f^0(x, \omega ) = x$$ f 0 ( x , ω ) = x and $$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$ f n ( x , ω ) = f ( f n - 1 ( x , ω ) , ω n ) for $$n \in {\mathbb {N}}$$ n ∈ N , and its weak limit $$\pi $$ π . We show that if $$\psi :X \rightarrow {\mathbb {R}}$$ ψ : X → R is continuous, then for every $$ x \in X $$ x ∈ X the sequence $$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$ 1 n ∑ k = 1 n ψ ( f k ( x , · ) ) n ∈ N converges almost surely to $$\int _X\psi d\pi $$ ∫ X ψ d π . In fact, we are focusing on the case where the metric space is complete and separable.

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann
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