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.
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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