Almost Everywhere Pointwise Convergence

Last updated Nov 1, 2022

Suppose $(X, \mathcal{M}, \mu)$ is a Measure Space and $(Y, \tau)$ is a Topological Space. Suppose $(f_{n})$ is a Sequence of Functions from $X$ to $Y$ and suppose $f: X \to Y$. If there exists a set $N \subset X$ so that

1. $\mu(N) = 0$
2. $\lim\limits_{n \to \infty} f_{n}(x) = f(x)$ for all $x \in N^{C}$.

then we say $f_{n}$ converges pointwise almost everywhere to $f$.