Almost Everywhere Convergence for Sequences of Continuous Functions
K. Schrader, S. Umamaheswaram · Proceedings of the American Mathematical Society · 1975
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 $n e m$. 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 $y$.