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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Almost everywhere convergence for sequences of continuous functions


Authors: K. Schrader and S. Umamaheswaram
Journal: Proc. Amer. Math. Soc. 47 (1975), 387-392
MSC: Primary 40A05
MathSciNet review: 0352773
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Abstract: The main result in this paper is the following theorem.

Theorem 1.1. Let $ \{ yk\} ,yk:I \to R$, be a sequence of continuous real valued functions defined on the bounded interval $ I$. Let $ {D_{kj}} = \{ x:x \in I,\vert yk(x) - yj(x)\vert > 0\} = { \cup _n}{I_{kjn}}$, where each $ {I_{kjn}}$ is a relatively open subinterval of $ I$ and $ {I_{kjn}} \cap {I_{kjm}} = \phi $ for $ n \ne m$. Assume there exists a function $ \phi ,\phi :(0, + \infty ) \to (0, + \infty )$, such that $ {\lim _{r \to {0^ + }}}(\phi (r)/r) = + \infty ,\phi $ is a nondecreasing function of $ r$ and

$\displaystyle \sum\limits_n {\phi (\mu ({I_{kjn}}))\mathop {\sup }\limits_{x\in {I_{kjn}}} \vert{y_k}(x) - {y_j}(x){\vert^p} \leq M} $

for all $ k,j$ sufficiently large, where $ \mu $ is Lebesgue measure and $ 0 < p < + \infty $. Then $ \{ {y_k}\} $ contains a subsequence which converges almost everywhere to a Lebesgue measurable function $ y$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0352773-X
PII: S 0002-9939(1975)0352773-X
Keywords: Sequences of functions, convergent subsequences, almost everywhere convergence
Article copyright: © Copyright 1975 American Mathematical Society