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

DOI:
https://doi.org/10.1090/S0002-9939-1975-0352773-X

MathSciNet review:
0352773

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Abstract: The main result in this paper is the following theorem.

**Theorem 1.1.** *Let , be a sequence of continuous real valued functions defined on the bounded interval . Let , where each is a relatively open subinterval of and for . Assume there exists a function , such that is a nondecreasing function of and*

*for all sufficiently large, where is Lebesgue measure and . Then contains a subsequence which converges almost everywhere to a Lebesgue measurable function*.

**[1]**F. Ramsey,*On a problem of formal logic*, Proc. London Math. Soc. (2)**30**(1930), 264-286.**[2]**-,*The foundations of mathematics*, Harcourt-Brace, New York, 1931.**[3]**K. Schrader,*A generalization of the Helly selection theorem*, Bull. Amer. Math. Soc.**78**(1972), 415-419. MR**45**#8788. MR**0299740 (45:8788)****[4]**-,*A pointwise convergence theorem for sequences of continuous functions*, Trans. Amer. Math. Soc.**159**(1971), 155-163. MR**43**#6621. MR**0280902 (43:6621)**

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0352773-X

Keywords:
Sequences of functions,
convergent subsequences,
almost everywhere convergence

Article copyright:
© Copyright 1975
American Mathematical Society