Inner measure

An inner measure is a function defined on all subsets of a given set, taking values in the extended real number system. It can be thought of as a counterpart to an outer measure. Whereas outer measures define the "size" of a set via a minimal covering set (from the outside), inner measures define the size of a set by approximating it with a maximal subset (from the inside). Although it is possible to develop measure theory with outer measures alone -- and many modern textbooks do just this -- Henri Lebesgue, in his 1902 thesis "Intégrale, longueur, aire" ("Integral, Length, Area"), used both inner measures and outer measures to construct the foundations of measure theory. It was not until Constantin Carathéodory made further contributions to the development of measure theory that inner measures were found to be redundant and an alternative notion of a "measurable" set was found.

Definition

Let $A\subseteq \mathbb {R}$ be an open set, and define a function $\mu :2^{\mathbb {R} }\rightarrow [0,\infty ]$ as follows. If $A=\emptyset$, then $\mu (A)=0$, and if $A$ is unbounded, then $\mu (A)=\infty$. Otherwise, if $A$ is open, then it can be written as a disjoint union of open intervals. Define $\mu (I)=b-a$ for any open interval $I=(a,b)$, and

$\mu (A)=\sum _{k=1}^{\infty }\mu (I_{k})$

whenever $A$ is a disjoint union of open intervals, $A=\cup _{k=1}^{\infty }I_{k}$.

It remains to show how $\mu$ is defined on closed, bounded subsets of $\mathbb {R}$ (equivalently, compact subsets of $\mathbb {R}$, by the Heine-Borel theorem). Let $B\subseteq \mathbb {R}$ be compact, and suppose $[a,b]$ is the smallest closed interval containing $B$. Define

$\mu (B)=b-a-\mu ((a,b)\setminus B).$

In other words, $\mu (B)$ is the measure of the smallest interval containing $B$, minus the measure of the complement of $B$ (which is an open set, and hence defined).

It is worth noting that this implies a particularly nice result, namely

$\mu ((a,b))=\mu (B)+\mu ((a,b)\setminus B).$

Outer Measures

The Lebesgue outer measure $\mu ^{*}$ is usually defined in terms of open intervals as

$\mu ^{*}(A)=\inf \left\{\sum _{i=1}^{\infty }|b_{i}-a_{i}|:A\subseteq \bigcup _{i=1}^{\infty }(a_{i},b_{i})\right\}.$

Using the function $\mu$ defined above, this can be rewritten in terms of general open sets as

$\mu ^{*}(A)=\inf\{\mu (G):A\subseteq G,G{\text{ open}}\}.$

Inner Measures

The Lebesgue inner measure of an arbitrary subset $A\subseteq \mathbb {R}$ is defined as

$\mu _{*}(A)=\sup\{\mu (F):F\subseteq A,F{\text{ compact}}\}.$

Since $E$ may not contain any intervals (e.g. in the case of the singleton $\{x\}$), there is no corresponding way to write this definition using open intervals. However, it is possible to define the Lebesgue inner measure in terms of the Lebesgue outer measure, as follows.

$\mu _{*}(A)=b-a-\mu ^{*}([a,b]\setminus A).$

Measurable Sets

Both inner measures and outer measures have one glaring flaw, namely that $\mu ^{*}(A_{1}\cup A_{2})$ need not be equal to $\mu ^{*}(A_{1})+\mu ^{*}(A_{2})$ (and equivalently for $\mu _{*}$). However, if one restricts to sets where $\mu ^{*}(A)=\mu _{*}(A)$, then countable additivity does hold. Hence, $A\subseteq \mathbb {R}$ is said to be a Lebesgue-measurable set if $\mu ^{*}(A)=\mu _{*}(A)$. The collection of Lebesgue-measurable sets forms a $\sigma$-algebra, denoted

${\mathcal {A}}=\{A\subseteq \mathbb {R} :\mu ^{*}(A)=\mu _{*}(A)\}.$

The Lebesgue measure $\lambda :{\mathcal {A}}\rightarrow [0,\infty ]$ can then be defined by $\lambda (A)=\mu ^{*}(A)$ for all $A\in {\mathcal {A}}$.

These definitions differ from those in common use today, but they agree with Henri Lebesgue's usage in his 1902 thesis. Later, Carathéodory removed the need for inner measures and defined a $\mu ^{*}$-measurable set to be any set $A\subseteq \mathbb {R}$ that satisfies

$\mu ^{*}(E)=\mu ^{*}(E\cap A)+\mu ^{*}(E\setminus A)$

for all $E\subseteq \mathbb {R}$.