outer measure

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

1.

$\mu^{\ast}(\emptyset)=0$ .
2.

If $A\subset B$ are subsets in $X$ , then $\mu^{\ast}(A)\leq\mu^{\ast}(B)$ .
3.

If $\{A_{i}\}$ is a countable collection of subsets of $X$ , then

$\mu^{\ast}(\bigcup_{i}A_{i})\leq\sum_{i}\mu^{\ast}(A_{i}).$

Here, we can make two remarks. First, from (1) and (2), it follows that $\mu^{\ast}$ is a positive function on $\mathcal{P}(X)$ . 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.

References

1 A. Mukherjea, K. Pothoven, Real and Functional analysis, Plenum press, 1978.
2 A. Friedman, Foundations of Modern Analysis, Dover publications, 1982.
3 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.

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