\begin{definition}[Definition:Convergence Almost Everywhere]
Let $\struct {X, \Sigma, \mu}$ be a measure space. 
Let $D \in \Sigma$.
Let $f: D \to \R$ be a $\Sigma$-measurable function.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n: D \to \R$.
Then $\sequence {f_n}_{n \mathop \in \N}$ is said to '''converge almost everywhere''' (or '''converge a.e.''') on $D$ to $f$ {{iff}}:
:$\map \mu {\set {x \in D : \sequence {\map {f_n} x}_{n \mathop \in \N} \text { does not converge to } \map f x} } = 0$
and we write $f_n \stackrel{a.e.} \to f$.
In other words, the sequence of functions converges pointwise outside of a $\mu$-null set.
\end{definition}