Definition:Almost Uniform Convergence

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Definition

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 $D$ if and only if:

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 $D \setminus E_\epsilon$.


Also see

A partial converse to this result is given by Egorov's Theorem.


Relations to Other Modes of Convergence

Convergence a.u. is weaker than uniform convergence.

Convergence a.u. also implies convergence in measure.

Sources