Inner measure

An inner measure is a set function

$$\backslash varphi\; :\; 2^X\; \backslash to\; [0,\; \backslash infty],$$

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, $$\backslash varphi(\backslash varnothing)\; =\; 0$$
• Superadditive: For any disjoint sets $A$ and $B,$ $$\backslash varphi(A\; \backslash cup\; B)\; \backslash geq\; \backslash varphi(A)\; +\; \backslash varphi(B).$$
• Limits of decreasing towers: For any sequence $A\_1,\; A\_2,\; \backslash ldots$ of sets such that $A\_j\; \backslash supseteq\; A\_\{j+1\}$ for each $j$ and $$\backslash varphi\; \backslash left(\backslash bigcap\_\{j=1\}^\backslash infty\; A\_j\backslash right)\; =\; \backslash lim\_\{j\; \backslash to\; \backslash infty\}\; \backslash varphi(A\_j)$$
• If the measure is not finite, that is, if there exist sets $A$ with $\backslash varphi(A)\; =\; \backslash infty$, then this infinity must be approached. More precisely, if $\backslash varphi(A)\; =\; \backslash infty$ for a set $A$ then for every positive real number $r,$ there exists some $B\; \backslash subseteq\; A$ such that

Let $\backslash Sigma$ be a σ-algebra over a set $X$ and $\backslash mu$ be a measure on $\backslash Sigma.$

Then the inner measure $\backslash mu\_*$ induced by $\backslash mu$ is defined by

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

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

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

{{main|Complete measure}}

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If $\backslash mu$ is a finite measure defined on a σ-algebra $\backslash Sigma$ over $X$ and $\backslash mu^*$ and $\backslash mu\_*$ are corresponding induced outer and inner measures, then the sets $T\; \backslash in\; 2^X$ such that $\backslash mu\_*(T)\; =\; \backslash mu^*(T)$ form a σ-algebra $\backslash hat\; \backslash Sigma$ with $\backslash Sigma\backslash subseteq\backslash hat\backslash Sigma$.Halmos 1950, § 14, Theorem F

The set function $\backslash hat\backslash mu$ defined by

$$\backslash hat\backslash mu(T)\; =\; \backslash mu^*(T)\; =\; \backslash mu\_*(T)$$

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

• {{annotated link|Lebesgue measurable set}}

{{reflist}}

• 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)

{{Measure theory}}

Category:Measures (measure theory)