construction of outer measures

The following theorem is used in measure theory to construct outer measures (http://planetmath.org/OuterMeasure2) on a set $X$, starting with a non-negative function on a collection of subsets of $X$. For example, if we take $X$ to be the real numbers, $\mathcal{C}$ to be the collection of bounded open intervals of $\mathbb{R}$ and define $p$ by $p((a,b))=b-a$ for real numbers $a, then the Lebesgue outer measure is obtained.

Theorem.

Let $X$ be a set, $\mathcal{C}$ be a family of subsets of $X$ containing the empty set and $p\colon\mathcal{C}\rightarrow\mathbb{R}\cup\{\infty\}$ be a function satisfying $p(\emptyset)=0$. Then the function $\mu^{*}\colon\mathcal{P}(X)\rightarrow\mathbb{R}\cup\{\infty\}$ defined by

 $\mu^{*}(A)=\inf\left\{\sum_{i=1}^{\infty}p(A_{i}):A_{i}\in\mathcal{C},\ A% \subseteq\bigcup_{i=1}^{\infty}A_{i}\right\}$ (1)

is an outer measure.

Proof.

The definition of $\mu^{*}$ immediately gives $\mu^{*}(A)\leq\mu^{*}(B)$ for sets $A\subseteq B$, and if $A=\emptyset$ then we can take $A_{i}=\emptyset$ in (1) to obtain $\mu^{*}(\emptyset)\leq\sum_{i}p(\emptyset)=0$, giving $\mu^{*}(\emptyset)=0$. Only the countable subadditivity of $\mu^{*}$ remains to be shown. That is, if $A_{i}$ is a sequence in $\mathcal{P}(X)$ then

 $\mu^{*}\left(\bigcup_{i}A_{i}\right)\leq\sum_{i}\mu^{*}(A_{i}).$ (2)

To prove this inequality, we may restrict to the case where $\mu^{*}(A_{i})<\infty$ for each $i$ so that, choosing any $\epsilon>0$, equation (1) says that there exists a sequence $A_{i,j}\in\mathcal{C}$ such that $A_{i}\subseteq\bigcup_{j}A_{i,j}$ and,

 $\sum_{j=1}^{\infty}p(A_{i,j})\leq\mu^{*}(A_{i})+2^{-i}\epsilon.$

As $\bigcup_{i}A_{i}\subseteq\bigcup_{i,j}A_{i,j}$, equation (1) defining $\mu^{*}$ gives

 $\mu^{*}\left(\bigcup_{i}A_{i}\right)\leq\sum_{i,j}p(A_{i,j})=\sum_{i}\sum_{j}p% (A_{i,j})\leq\sum_{i}(\mu^{*}(A_{i})+2^{-i}\epsilon)=\sum_{i}\mu^{*}(A_{i})+\epsilon.$

As $\epsilon>0$ is arbitrary, this proves subadditivity (2). ∎

Although this result is rather general, placing few restrictions on the function $p$, there is no guarantee that the outer measure $\mu^{*}$ will agree with $p$ for the sets in $\mathcal{C}$ nor that $\mathcal{C}$ will consist of $\mu^{*}$-measurable (http://planetmath.org/CaratheodorysLemma) sets.

For example, if $X=\mathbb{R}$, $\mathcal{C}$ consists of the bounded open intervals, and $p((a,b))=(b-a)^{2}$ for real numbers $a, then $\mu^{*}((a,b))=0\not=p((a,b))$.

Alternatively if $p((a,b))=\sqrt{b-a}$ for all $a then it follows that $\mu^{*}((a,b))=\sqrt{b-a}$ so

 $\mu^{*}((0,1))+\mu^{*}([1,2))=1+1\not=\mu^{*}((0,2))=\sqrt{2},$

and $(0,1)$ is not $\mu^{*}$-measurable.

Title construction of outer measures ConstructionOfOuterMeasures 2013-03-22 18:33:17 2013-03-22 18:33:17 gel (22282) gel (22282) 10 gel (22282) Theorem msc 28A12 OuterMeasure LebesgueOuterMeasure CaratheodorysLemma