Definition:Convergence Almost Everywhere
Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.
Let $D \in \Sigma$.
Let $f: D \to \R$ be a $\Sigma$-measurable function.
Let $\left({f_n}\right)_{n \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n: D \to \R$.
Then $\left({f_n}\right)_{n \in \N}$ is said to converge almost everywhere (or converge a.e.) on $D$ to $f$ if and only if:
- $\mu \left({ \left\{{x \in D : f_n \left({x}\right) \text{ does not converge to } f \left({x}\right) }\right\} }\right) = 0$
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.
Relations to Other Modes of Convergence
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Convergence a.e. is also implied by pointwise convergence.
Convergence a.e. implies convergence in measure.