Definition:Convergence Almost Everywhere
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Definition
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 $f$ if and only if:
- $\map \mu {\set {x \in D : \sequence {\map {f_n} x}_{n \mathop \in \N} \text { does not converge to } \map f x} } = 0$
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.
Also see
- Convergence a.u. Implies Convergence a.e.. A partial converse to this result is given by Egorov's Theorem.
- Pointwise Convergence implies Convergence Almost Everywhere
- Convergence Almost Everywhere with respect to Finite Measure implies Convergence in Measure
- Set of Points at which Sequence of Measurable Functions does not Converge to Given Measurable Function is Measurable: shows that the set $\set {x \in D : \sequence {\map {f_n} x}_{n \mathop \in \N} \text { does not converge to } \map f x}$ is always $\Sigma$-measurable.
- Results about convergence almost everywhere can be found here.