Given a measurable subset $A$ of $[0, 1]$, a sequence of functions $f_n: [0, 1] \to \mathbb R$ is said to be equi-Lebesgue continuous on $A$ if for every $x \in A$, and $\varepsilon > 0$, there exists some $\delta > 0$ such that for all $0 < r < \delta$, we have

$$\frac{1}{2r} \int_{B_r (x)} |f_n (x) - f_n (y)| \, dy < \varepsilon$$

for all $n \in \mathbb N$.

Let $f_n: [0, 1] \to \mathbb R$ be a sequence of functions equibounded in $L^\infty$, that is, $\sup_{n \in \mathbb N} \lvert \lvert f_n \rvert \rvert_{L^\infty} < \infty$. Suppose further that there exists a subset $E$ of $[0, 1]$ of measure $1$ such that $f_n$ are equi-Lebesgue continuous on $E$.

Question: Does there exist a subsequence $f_{n_k}$ of $f$ converging a.e.?

Given a measurable subset $A$ of $[0, 1]$, a sequence of functions $f_n: [0, 1] \to \mathbb R$ is said to be equi-Lebesgue continuous on $A$ if for every $x \in A$, and $\varepsilon > 0$, there exists some $\delta > 0$ such that for all $0 < r < \delta$, we have

$$\frac{1}{2r} \int_{B_r (x)} \lvert f_n (x) - f_n (y)\rvert \, dy < \varepsilon$$

for all $n \in \mathbb N$.

Let $f_n: [0, 1] \to \mathbb R$ be a sequence of functions equibounded in $L^\infty$, that is, $\sup_{n \in \mathbb N} \lVert f_n \rVert_{L^\infty} < \infty$. Suppose further that there exists a subset $E$ of $[0, 1]$ of measure $1$ such that $f_n$ are equi-Lebesgue continuous on $E$.

Question: Does there exist a subsequence $f_{n_k}$ of $f$ converging a.e.?