Almost everywhere convergence for sequences of continuous functions
HTML articles powered by AMS MathViewer
- by K. Schrader and S. Umamaheswaram PDF
- Proc. Amer. Math. Soc. 47 (1975), 387-392 Request permission
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 ${D_{kj}} = \{ x:x \in I,|yk(x) - yj(x)| > 0\} = { \cup _n}{I_{kjn}}$, 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 \[ \sum \limits _n {\phi (\mu ({I_{kjn}}))\sup \limits _{x\in {I_{kjn}}} |{y_k}(x) - {y_j}(x){|^p} \leq M} \] 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 $y$.References
-
F. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286.
—, The foundations of mathematics, Harcourt-Brace, New York, 1931.
- Keith Schrader, A generalization of the Helly selection theorem, Bull. Amer. Math. Soc. 78 (1972), 415–419. MR 299740, DOI 10.1090/S0002-9904-1972-12923-9
- K. Schrader, A pointwise convergence theorem for sequences of continuous functions, Trans. Amer. Math. Soc. 159 (1971), 155–163. MR 280902, DOI 10.1090/S0002-9947-1971-0280902-6
Additional Information
- © Copyright 1975 American Mathematical Society
- 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