outer measure

Definition [1, 2, 3]Let $X$ be a set, and let $𝒫(X)$ be thepower set of $X$. An outer measure on $X$ is a function${\mu }^{\ast }:𝒫(X)\to [0,\mathrm{\infty }]$ satisfying the properties

1. ${\mu }^{\ast }(\mathrm{\varnothing })=0$.

2. If $A\subset B$ are subsets in $X$, then ${\mu }^{\ast }(A)\le {\mu }^{\ast }(B)$.

3. If $\{A_i\}$ is a countable collection of subsets of $X$,then

   $${\mu }^{\ast }(\bigcup _i{A}_i)\le \sum _i{\mu }^{\ast }(A_i).$$

Here, we can make two remarks. First, from (1) and (2), it followsthat ${\mu }^{\ast }$ is a positive function on $𝒫(X)$. Second,property (3) also holds for any finite collection of subsets sincewe can always append an infinite sequence of empty sets tosuch a collection.