Inner measure
# Inner measure

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.

## Definition

An inner measure is a set function

$\varphi :2^{X}\to [0,\infty ],$

defined on all subsets of a set $X,$ that satisfies the following conditions:

* Null empty set: The empty set has zero inner measure (see also: measure zero); that is, $\varphi (\varnothing )=0$
* Superadditive: For any disjoint sets $A$ and $B,$ $\varphi (A\cup B)\geq \varphi (A)+\varphi (B).$
* Limits of decreasing towers: For any sequence $A_{1},A_{2},\ldots$ of sets such that $A_{j}\supseteq A_{j+1}$ for each $j$ and $\varphi (A_{1})<\infty$ $\varphi \left(\bigcap _{j=1}^{\infty }A_{j}\right)=\lim _{j\to \infty }\varphi (A_{j})$
* Infinity must be approached: If $\varphi (A)=\infty$ for a set $A$ then for every positive real number $r,$ there exists some $B\subseteq A$ such that $r\leq \varphi (B)<\infty .$

## The inner measure induced by a measure

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

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

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

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

## Measure completion

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

The set function ${\hat {\mu }}$ defined by

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

for all $T\in {\hat {\Sigma }}$ is a measure on ${\hat {\Sigma }}$ known as the completion of $\mu .$

