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
![{\displaystyle \varphi :2^{X}\to [0,\infty ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1682da42090dc091d1599a2ce9ecb44fc4fdbf40)
defined on all
subsets of a set
![{\displaystyle X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df)
that satisfies the following conditions:
- Null empty set: The empty set has zero inner measure (see also: measure zero); that is,
![{\displaystyle \varphi (\varnothing )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84a9d2472389060e8c0ead7bd40e211333ee62a8)
- Superadditive: For any disjoint sets
and
![{\displaystyle \varphi (A\cup B)\geq \varphi (A)+\varphi (B).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b44e31da2ea0d7e43cfc8f1e8fb5a2bf1ce4bfd9)
- Limits of decreasing towers: For any sequence
of sets such that
for each
and
![{\displaystyle \varphi \left(\bigcap _{j=1}^{\infty }A_{j}\right)=\lim _{j\to \infty }\varphi (A_{j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/828385901511898a98a906cc420632bc03fc3337)
- Infinity must be approached: If
for a set
then for every positive real number
there exists some
such that ![{\displaystyle r\leq \varphi (B)<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f79909f243880b44c88c9baf90a06676cea19e)
The inner measure induced by a measure
Let
be a σ-algebra over a set
and
be a measure on
Then the inner measure
induced by
is defined by
![{\displaystyle \mu _{*}(T)=\sup\{\mu (S):S\in \Sigma {\text{ and }}S\subseteq T\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/410e358c002186dd2592ebc916224c9ffb8b26b0)
Essentially
gives a lower bound of the size of any set by ensuring it is at least as big as the
-measure of any of its
-measurable subsets. Even though the set function
is usually not a measure,
shares the following properties with measures:
![{\displaystyle \mu _{*}(\varnothing )=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d3cca97721572af501972c3d860c4dc9293b55)
is non-negative,
- If
then ![{\displaystyle \mu _{*}(E)\leq \mu _{*}(F).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/255f583d5300fd1e2f6319931829ec287b8cee1b)
Measure completion
Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If
is a finite measure defined on a σ-algebra
over
and
and
are corresponding induced outer and inner measures, then the sets
such that
form a σ-algebra
with
.[1]
The set function
defined by
![{\displaystyle {\hat {\mu }}(T)=\mu ^{*}(T)=\mu _{*}(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e78c973492f866a4491ea79f094245b263a34f3f)
for all
![{\displaystyle T\in {\hat {\Sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0c1326adff25e47bdbebb2d5f2feac2c6503c5)
is a measure on
![{\displaystyle {\hat {\Sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bf20d3001fc43165078ae33179fce02a553c8d)
known as the completion of
See also
References
- ^ Halmos 1950, § 14, Theorem F
- 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)