Probabilistic norms and convergence of random variables

Lafuerza-Guillen, Bernardo; Sempi, Carlo

doi:10.1016/s0022-247x(02)00577-2

Cited by 23 publications

(9 citation statements)

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“…Following K. Menger's idea, A. N. Serstnev presented the notion of a probabilistic normed space (briefly, a P N space) in 1962, then in 1993 C. Alsina, B. Schweizer and A. Sklar redefined P N spaces in a more general way in [2] and in 1997 they and C. Sempi presented the notion of a probabilistic inner product space (briefly, a P IP space) in [3]. P N spaces are usually endowed with a natural topology, called the (ε, λ)-topology, so that they are metrizable linear topological spaces under a mild condition [4], see [5][6][7] for the closely related studies of P N spaces. Since P N spaces are rarely locally convex spaces, for example, Menger P N spaces under a t-norm other than the t-norm Min are not locally convex spaces in general, even they do not admit a nontrival continuous linear functional, and so the theory of traditional conjugate spaces universally fails to serve for the deep development of P N spaces.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Guo

2008

Sci. China Ser. A-Math.

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Let (Ω, A, μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S, X ) a random normed space over K with base (Ω, A, μ). Denote the support of (S, X ) by E, namely E is the essential supremum of the set {A ∈ A : there exists an element p in S such that Xp(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S * (1) = {f ∈ S * : X * f 1} of the random conjugate space (S * , X * ) of (S, X ) is compact under the random weak star topology on (S * , X * ) iff E A=: {E A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {An : n ∈ N } of at most countably many μ-atoms from E A such that E = ∞ n=1 An and for each element F in E A, there is an H in the σ-algebra generated by {An : n ∈ N } satisfying μ(F H) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S, X ) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S : Xp 1} of (S, X ) is compact under the random weak topology on (S, X ) iff both (S, X ) is random reflexive and E A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.Keywords: random normed module, random reflexivity, random weak star compactness, random weak compactness, stratification structure MSC(2000): 46A16, 46H25, 46B10, 54E45, 54E70

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Guo

2008

Sci. China Ser. A-Math.

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“…Such spaces were introduced by Sherwwod ( [46], [47], but see also [35, section 12.1]). The results that follow have been established in [16] and [20]. …”

Section: En Spacesmentioning

confidence: 80%

A Short and Partial History of Probabilistic Normed Spaces

Sempi

2006

MedJM

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“…Serstnev presented the notion of a probabilistic normed space (briefly, a P N space) in 1962, then in 1993 C. Alsina, B. Schweizer and A. Sklar redefined P N spaces in a more general way in [2] and in 1997 they and C. Sempi presented the notion of a probabilistic inner product space (briefly, a P I P space) in [3]. P N spaces are usually endowed with a natural topology, called the (ε, λ)-topology, so that they are metrizable linear topological spaces under a mild condition [4], see [5][6][7] for the closely related studies of P N spaces. Since P N spaces are rarely locally convex spaces, for example, Menger P N spaces under a t-norm other than the t-norm Min are not locally convex spaces in general, even they do not admit a nontrival continuous linear functional, and so the theory of traditional conjugate spaces universally fails to serve for the deep development of P N spaces.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

Guo

Shi

2011

Journal of Mathematical Analysis and Applications

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Probabilistic norms and convergence of random variables

Abstract: We prove that the probabilistic norms of suitable Probabilistic Normed spaces induce convergence in probability, LI' convergence and almost sure convergence. o 2003 Elsevier Science (USA). All rights reserved.

Cited by 23 publications

References 10 publications

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

A Short and Partial History of Probabilistic Normed Spaces

The algebraic structure of finitely generated L0(F,K)-modules and the Helly theorem in random normed modules

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