---- > [!definition] Definition. ([[outer measure]]) > Let $X$ be a set and $2^{X}$ its [[power set]]. An **outer measure** on $X$ is a function $\mu:2^{X} \to [0, \infty]$ such that: >- $\mu(\emptyset)=0$; >- ([[monotonic map|monotone]]) If $A, B \subset X$ with $A \subset B$, then $\mu(A) \leq \mu(B)$ . >- (Countable subadditivity) For arbitrary subsets $B_{1},B_{2},\dots$ of $X$, $\mu(\bigcup_{j=1} ^{\infty}B_{j}) \leq \sum_{j=1}^{\infty} \mu(B_{j})$. > Crucially, an outer measure need not be a [[measure]]: the three properties above do not guarantee countable additivity. (This is the case e.g. for the [[Lebesgue outer measure]] on $\mathbb{R}$). Any earnest [[measure]] $(\mu, \Sigma)$ on $X$ induces a canonical [[outer measure]] $\mu^{*}:2^{X}\to[0, \infty]$ as $\mu^{*}(E \in 2^{X}):= \inf \left\{ \mu(A) : A \in \Sigma, E \subset A \right\}.$ ---- #### ---- #### 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 > ```