---- > [!definition] Definition. ([[converges almost-everywhere]]) > Let $(X,\Sigma,\mu)$ be a [[measure|measure space]] and $Y$ a [[topological space]] (usually [[metrizable]]). We say a [[sequence]] of functions $(f_{n}:X \to Y)$ **converges almost-everywhere (a.e.)** to $f:X \to Y$ if there exists a set $N \subset X$ [[outer measure|with]] $\mu^{*}(N)=0$ such that $f_{n}(x) \to f(x)$ for all $x \not \in N$. This is a weakening of the notion of [[pointwise convergence]]. > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```