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 $\mu^\ast(A)\le \mu^\ast(B)$ .
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 $\mathcal{P}(X)$ . 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.

Bibliography

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.