# 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**.