Definition:Almost Uniform Convergence

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 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 known as

Almost uniform convergence is also known as convergence a.u.

Also see

• Convergence a.u. Implies Convergence a.e.

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

• Results about almost uniform convergence can be found here.

Relations to Other Modes of Convergence

Convergence a.u. is weaker than uniform convergence.

Convergence a.u. also implies convergence in measure.

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Sources

• 1981: G. de Barra: Measure Theory and Integration: $\S 7.2$: Almost Uniform Convergence: Definition $4$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): almost uniform convergence
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): almost uniform convergence