Definition:Convergence Almost Everywhere

Given a measure space $$(X, \Sigma, \mu)\ $$ and a sequence of $\Sigma$-measurable functions $$f_n:D\to\R$$ for $$D\in\Sigma$$, the sequence is said to converge almost everywhere (or converge a.e.) on $$D\ $$ to a function $$f\ $$ if
 * $$\mu(\{x\in D:f_n(x) \text{ does not converge to } f(x)\}) = 0$$,

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

In other words, the sequence of functions converges pointwise outside of a 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.