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 , \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 $A_j \supseteq A_$ for each $j$ and $\varphi(A_1) < \infty$ $\varphi \left(\bigcap_^\infty A_j\right) = \lim_ \varphi(A_j)$
* 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
$\mu_*(T) = \sup\.$

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 $\Sigma\subseteq\hat\Sigma$.Halmos 1950, § 14, Theorem F
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

*

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, (Chapter 7)

{{Measure theory

Measures (measure theory)