Inner measure explained

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$$\backslash varphi\; :\; 2^X\; \backslash to\; [0,\; \backslash infty],$$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, $$\backslash varphi(A\; \backslash cup\; B)\; \backslash geq\; \backslash varphi(A)\; +\; \backslash varphi(B).$$A _{1,} A _{2,} \ldots of sets such that A _{j} \supseteq A _{j+1} for each j and \varphi(A _{1)} < inf ty $$\backslash varphi\; \backslash left(\backslash bigcap\_^\backslash infty\; A\_j\backslash right)\; =\; \backslash lim\_\; \backslash varphi(A\_j)$$

• If the measure is not finite, that is, if there exist sets

A with \varphi(A) = inf ty , then this infinity must be approached. More precisely, if \varphi(A) = inf ty for a set A then for every positive real number r, there exists some B \subseteq A such that

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$$\backslash mu\_*(T)\; =\; \backslash sup\backslash .$$

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,

1. If

E \subseteq F then \mu _{*(E)} \leq \mu _{*(F).}

Measure completion

See 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 $\backslash hat\; \backslash Sigma$ with $\backslash Sigma\backslash subseteq\backslash hat\backslash Sigma$.^[1] The set function \hat\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 \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, (Chapter 7)

Notes and References

1. Halmos 1950, § 14, Theorem F