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 $\varphi (A_{1})<\infty$
  $$\varphi \left(\bigcap _{j=1}^{\infty }A_{j}\right)=\lim _{j\to \infty }\varphi (A_{j})$$
• If the measure is not finite, that is, if there exist sets $A$ with $\varphi (A)=\infty$, then this infinity must be approached. More precisely, 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:

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

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 $\mu$ 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 $\mu .$

See also

• Lebesgue measurable set – Concept of area in any dimensionPages displaying short descriptions of redirect targets

References

• 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)