Definition:Convergence Almost Everywhere
From ProofWiki
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.
