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 $$\phi :2^X\to [0,\mathrm{\infty }],$$ 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, $$\phi (\varnothing )=0$$
• Superadditive: For any disjoint sets $A$ and $B,$ $$\phi (A\cup B)\ge \phi (A)+\phi (B).$$
• Limits of decreasing towers: For any sequence $A_1,A_2,\dots$ of sets such that $A_j\supseteq A_{j+1}$ for each $j$ and $\phi (A_1)<\mathrm{\infty }$ $$\phi \left(\bigcap _{j=1}^{\mathrm{\infty }}{A}_j\right)=\underset{j\to \mathrm{\infty }}{\lim }\phi (A_j)$$
• If the measure is not finite, that is, if there exist sets $A$ with $\phi (A)=\mathrm{\infty }$, then this infinity must be approached. More precisely, if $\phi (A)=\mathrm{\infty }$ for a set $A$ then for every positive real number $r,$ there exists some $B\subseteq A$ such that $$r\le \phi (B)<\mathrm{\infty }.$$

The inner measure induced by a measure

Let $\mathrm{\Sigma }$ be a σ-algebra over a set $X$ and $\mu$ be a measure on $\mathrm{\Sigma }.$ Then the inner measure ${\mu }_{*}$ induced by $\mu$ is defined by $${\mu }_{*}(T)=\sup \{\mu (S):S\in \mathrm{\Sigma }\text{ and }S\subseteq T\}.$$

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 $\mathrm{\Sigma }$-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 ${\mu }_{*}(E)\le {\mu }_{*}(F).$

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 $\mathrm{\Sigma }$ 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 $\hat{\mathrm{\Sigma }}$ with $\mathrm{\Sigma }\subseteq \hat{\mathrm{\Sigma }}$. The set function $\hat{\mu }$ defined by $$\hat{\mu }(T)={\mu }^{*}(T)={\mu }_{*}(T)$$ for all $T\in \hat{\mathrm{\Sigma }}$ is a measure on $\hat{\mathrm{\Sigma }}$ known as the completion of $\mu .$

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)