Relations between Lp-and pointwise convergence of families of functions indexed by the unit interval

Vaios Laschos, C Monch

Research output: Contribution to journalArticlepeer-review

Abstract

We construct a variety of mappings from the unit interval I into Lp([0,1]); 1 ≤ p > to generalize classical examples of Lp-converging equences of functions with simultaneous pointwise divergence. By es-tablishing relations between the regularity of the functions in the image of the mappings and the topology of I, we obtain examples which are Lp-continuous but exhibit discontinuity in a pointwise sense to different egrees. We conclude by proving a Lusin-type theorem, namely that if almost every function in the image is continuous, then we can remove a set of arbitrarily small measure from the index set I and establish pointwise continuity in the remainder.

Original languageEnglish
Pages (from-to)177-192
Number of pages16
JournalReal Analysis Exchange
Volume38
Issue number1
Publication statusPublished - 2013

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