# Forcing Divergence When the Supremum Is Not Integrable

Forcing Divergence When the Supremum Is Not Integrable If a sequence of functions (f n ) has an integrable supremum and converges almost everywhere, then an operator sequence (T n ) will yield a sequence (T n f n ) that converges almost everywhere too, under some very general assumptions about the sequence (T n ). However, if the supremum of (f n ) is not integrable then this can fail to be the case. It is shown that if a sequence of functions (f n )has a supremum sup n |f n | that is not integrable, then one can always construct a variety of sequences of positive contractions (T n ) such that lim sup |T n f n | = ∞ a.e. These operators can be conditional expectations with respect to an increasing sequence of finite σ-algebras, the conditional expectation with respect to one fixed σ-algebra, or averages with respect to a measure-preserving transformation. General discussion of these constructions, history of previous results of this type, and some open questions are also given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Forcing Divergence When the Supremum Is Not Integrable

, Volume 10 (2) – Apr 26, 2006
24 pages

/lp/springer_journal/forcing-divergence-when-the-supremum-is-not-integrable-h8SXsE077n
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-005-0024-z
Publisher site
See Article on Publisher Site

### Abstract

If a sequence of functions (f n ) has an integrable supremum and converges almost everywhere, then an operator sequence (T n ) will yield a sequence (T n f n ) that converges almost everywhere too, under some very general assumptions about the sequence (T n ). However, if the supremum of (f n ) is not integrable then this can fail to be the case. It is shown that if a sequence of functions (f n )has a supremum sup n |f n | that is not integrable, then one can always construct a variety of sequences of positive contractions (T n ) such that lim sup |T n f n | = ∞ a.e. These operators can be conditional expectations with respect to an increasing sequence of finite σ-algebras, the conditional expectation with respect to one fixed σ-algebra, or averages with respect to a measure-preserving transformation. General discussion of these constructions, history of previous results of this type, and some open questions are also given.

### Journal

PositivitySpringer Journals

Published: Apr 26, 2006

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