2009
DOI: 10.1515/dema-2013-0205
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Expectation in metric spaces and characterizations of Banach spaces

Abstract: We consider different definitions of expectation of random elements taking values in metric spaces. All such definitions are valid also in Banach spaces and in this case the results coincide with the Bochner integral. There may exist an isometry between considered metric space and some Banach space and in this case one can use the Bochner integral instead of expectation in metric space. We give some conditions which ensure existence of such isometry, for two different definitions of expectation in metric space. Show more

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Cited by 2 publications
(2 citation statements)
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“…Fréchet expectation is one of the examples. Moreover I. Molchanov and P. Teran mentioned that strong law of large numbers proved by K. T. Sturm for random elements taking values in global NPC spaces (see [13]) follows from their strong law of large numbers but this is true only in case when Lemma 1 holds but this case is not really interesting because then we have the following: Theorem 1 (see [2] …”
Section: Lemmamentioning
confidence: 96%
“…Fréchet expectation is one of the examples. Moreover I. Molchanov and P. Teran mentioned that strong law of large numbers proved by K. T. Sturm for random elements taking values in global NPC spaces (see [13]) follows from their strong law of large numbers but this is true only in case when Lemma 1 holds but this case is not really interesting because then we have the following: Theorem 1 (see [2] …”
Section: Lemmamentioning
confidence: 96%
“…The collection {X n , F n , n 1} is said to be martingale if X n is F n -measurable and E(X n+1 |F n ) = X n a.s. for all n 1. Thanks to Corollary 4.12 (1), it is easy to verify that if {X n , F n , n 1} is a martingale then { X n p a , F n , n 1} is a real-valued submartingale for a ∈ X, p 1 arbitrarily. The convergence of martingales will be established in proposition below.…”
Section: The Injection Of J Implies E([imentioning
confidence: 99%