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. 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

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

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

When working with subsets of $\mathbb {R}$, any open set can be written as a union of open intervals. Therefore, it is possible to define the outer measure in terms of general open sets, as

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

where $\mu (B)$ is defined in the intuitive way, i.e. $\mu (I)=b-a$ for any open interval $I=(a,b)$, and

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

whenever $B$ is a disjoint union of open intervals, $B=\cup _{k=1}^{\infty }I_{k}$. If $B$ is unbounded, then $\mu (B)=\infty$, and if $B=\emptyset$, then $\mu (B)=0$.