In mathematics, in particular in measure theory, an inner measure is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set.

The inner measure induced by a measure

Let ${\displaystyle \Sigma }$ be a σ-algebra over a set ${\displaystyle X}$ and ${\displaystyle \mu }$ be a measure on ${\displaystyle \Sigma .}$ Then the inner measure ${\displaystyle \mu _{*))$ induced by ${\displaystyle \mu }$ is defined by

$${\displaystyle \mu _{*}(T)=\sup\{\mu (S):S\in \Sigma {\text{ and ))S\subseteq T\}.}$$

Essentially ${\displaystyle \mu _{*))$ gives a lower bound of the size of any set by ensuring it is at least as big as the ${\displaystyle \mu }$-measure of any of its ${\displaystyle \Sigma }$-measurable subsets. Even though the set function ${\displaystyle \mu _{*))$ is usually not a measure, ${\displaystyle \mu _{*))$ shares the following properties with measures:

1. ${\displaystyle \mu _{*}(\varnothing )=0,}$
2. ${\displaystyle \mu _{*))$ is non-negative,
3. If ${\displaystyle E\subseteq F}$ then ${\displaystyle \mu _{*}(E)\leq \mu _{*}(F).}$

Measure completion

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If ${\displaystyle \mu }$ is a finite measure defined on a σ-algebra ${\displaystyle \Sigma }$ over ${\displaystyle X}$ and ${\displaystyle \mu ^{*))$ and ${\displaystyle \mu _{*))$ are corresponding induced outer and inner measures, then the sets ${\displaystyle T\in 2^{X))$ such that ${\displaystyle \mu _{*}(T)=\mu ^{*}(T)}$ form a σ-algebra ${\displaystyle {\hat {\Sigma ))}$ with ${\displaystyle \Sigma \subseteq {\hat {\Sigma ))}$.^[1] The set function ${\displaystyle {\hat {\mu ))}$ defined by

$${\displaystyle {\hat {\mu ))(T)=\mu ^{*}(T)=\mu _{*}(T)}$$

${\displaystyle T\in {\hat {\Sigma ))}$

${\displaystyle {\hat {\Sigma ))}$

${\displaystyle \mu .}$