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

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

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

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

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

Main article: Complete measure

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

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

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

See also

• Lebesgue measurable set

References

1. Halmos 1950, § 14, Theorem F

• Halmos, Paul R., Measure Theory, D. Van Nostrand Company, Inc., 1950, pp. 58.
• A. N. Kolmogorov & S. V. Fomin, translated by Richard A. Silverman, Introductory Real Analysis, Dover Publications, New York, 1970, ISBN   0-486-61226-0 (Chapter 7)