Derivation, -bounded martingales and covering conditions

Author:
Michel Talagrand

Journal:
Trans. Amer. Math. Soc. **293** (1986), 257-291

MSC:
Primary 28A15; Secondary 46G05, 60G42

MathSciNet review:
814922

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a complete probability space. Let be a directed family of sub--algebras of . Let be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of -bounded martingales. We do not assume that either or satisfy the condition. We show that when satisfies condition Exp, that is when there exists an such that for each , the essential convergence of -bounded martingales is equivalent to the classical covering condition . This covers in particular the classical case . The growth condition Exp on cannot be relaxed. When contains a countable cofinite set, we show that the essential convergence of -bounded martingales is equivalent to a covering condition (that is weaker than ). When fails condition Exp, condition is optimal. Roughly speaking, in the case of -bounded martingales, condition means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where does not contain a countable cofinal set and fails condition Exp. In this case, it seems impossible to characterize the essential convergence of -bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable -bounded martingales, but not all -bounded martingales, converge essentially. Similar results are also established in the derivation setting.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0814922-8

Keywords:
Derivation,
martingales,
covering conditions,
Orlicz spaces

Article copyright:
© Copyright 1986
American Mathematical Society