\begin{definition}[Definition:Almost Uniform Convergence]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $D \in \Sigma$.
Let $\sequence {f_n}_{n \mathop \in \N}, f_n: D \to \R$ be a sequence of $\Sigma$-measurable functions.
Then $\sequence {f_n}_{n \mathop \in \N}$ is said to '''converge almost uniformly''' (or '''converge a.u.''') on $D$ {{iff}}:
:For all $\epsilon > 0$, there is a measurable subset $E_\epsilon \subseteq D$ of $D$ such that:
::$(1): \quad \map \mu {E_\epsilon} < \epsilon$;
::$(2): \quad \sequence {f_n}_{n \mathop \in \N}$ converges uniformly to $f$ on $D \setminus E_\epsilon$.
\end{definition}