On characterization of a.s. convergence of all conditionings for sequences of random variables

We prove that in a non-atomic probability space, for a sequence of any integrable r.v. ( ) we have the following equivalence: E( |U) → 0 a.s. for any σ-field U of events iff → 0 a.s. and E sup | | < ∞.

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Veröffentlicht in:	Demonstratio mathematica 2012-06, Vol.45 (2), p.455-462
1. Verfasser:	Paszkiewicz, Adam
Format:	Artikel
Sprache:	eng
Schlagworte:	28A20 40A30 almost everywhere convergences conditional expectations
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container_title	Demonstratio mathematica
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description	We prove that in a non-atomic probability space, for a sequence of any integrable r.v. ( ) we have the following equivalence: E( \|U) → 0 a.s. for any σ-field U of events iff → 0 a.s. and E sup \| \| < ∞.
doi_str_mv	10.1515/dema-2013-0376
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subjects	28A20 40A30 almost everywhere convergences conditional expectations
title	On characterization of a.s. convergence of all conditionings for sequences of random variables
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