Definition:Convergence Almost Everywhere

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $D \in \Sigma$.

Let $f: D \to \R$ be a $\Sigma$-measurable function.

Let $\left({f_n}\right)_{n \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n: D \to \R$.

Then $\left({f_n}\right)_{n \in \N}$ is said to converge almost everywhere (or converge a.e.) on $D$ to $f$ iff:


 * $\mu \left({ \left\{{x \in D : f_n \left({x}\right) \text{ does not converge to } f \left({x}\right) }\right\} }\right) = 0$

and we write $f_n \stackrel{a.e.}{\to} f$.

In other words, the sequence of functions converges pointwise outside of a $\mu$-null set.

Relations to Other Modes of Convergence
Convergence a.e. is implied by uniform convergence a.e. (proof here). A partial converse to this result is given by Egorov's Theorem.

Convergence a.e. is also implied by pointwise convergence.

Convergence a.e. implies convergence in measure.