outer measure

Definition [1, 2, 3] Let $X$ be a set, and let $𝒫(X)$ be the power 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 follows that ${\mu }^{\ast }$ is a positive function on $𝒫(X)$. Second, property (3) also holds for any finite collection of subsets since we can always append an infinite sequence of empty sets to such a collection.

Title	outer measure
Canonical name	OuterMeasure
Date of creation	2013-03-22 13:45:20
Last modified on	2013-03-22 13:45:20
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Definition
Classification	msc 60A10
Classification	msc 28A10
Related topic	CaratheodorysExtensionTheorem
Related topic	CaratheodorysLemma
Related topic	ProofOfCaratheodorysExtensionTheorem