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. Positivity Springer Journals

Forcing Divergence When the Supremum Is Not Integrable

Loading next page...
Copyright © 2006 by Birkhäuser Verlag, Basel
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
Publisher site
See Article on Publisher Site


You’re reading a free preview. Subscribe to read the entire article.

DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches


Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.



billed annually
Start Free Trial

14-day Free Trial