Lerche–Newberger sum rule: Difference between revisions
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In [[mathematics]], in particular in [[measure theory]], an '''inner measure''' is a [[Function (mathematics)|function]] on the [[Set (mathematics)|set]] of all subsets of a given set, with values in the [[extended real line|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 function | |||
:<math>\varphi: 2^X \rightarrow [0, \infty], </math> | |||
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]]''). | |||
:: <math> \varphi(\varnothing) = 0</math> | |||
* [[superadditivity|Superadditive]]: For any disjoint sets A and B, | |||
:: <math> \varphi( A \cup B) \geq \varphi(A) + \varphi( B ).</math> | |||
* Limits of decreasing towers: For any sequence {''A''<sub>''j''</sub>} of sets such that <math> A_j \supseteq A_{j+1}</math> for each ''j'' and <math> \varphi(A_1) < \infty</math> | |||
:: <math> \varphi \left(\bigcap_{j=1}^\infty A_j\right) = \lim_{j \to \infty} \varphi(A_j)</math> | |||
* Infinity must be approached: If <math>\varphi(A) = \infty </math> for a set A then for every positive number c, there exists a B which is a subset of A such that, | |||
:: <math> c \leq \varphi( B) <\infty</math> | |||
== The inner measure induced by a measure == | |||
Let ''Σ'' be a σ-algebra over a set ''X'' and ''μ'' be a [[Measure (mathematics)|measure]] on ''Σ''. | |||
Then the inner measure ''μ''<sub>*</sub> induced by ''μ'' is defined by | |||
:<math>\mu_*(T)=\sup\{\mu(S):S\in\Sigma\text{ and }S\subseteq T\}.</math> | |||
Essentially ''μ''<sub>*</sub> gives a lower bound of the size of any set by ensuring it is at least as big as the ''μ''-measure of any of its ''Σ''-measurable subsets. Even though the set function ''μ''<sub>*</sub> is usually not a measure, ''μ''<sub>*</sub> shares the following properties with measures: | |||
:# ''μ''<sub>*</sub>(∅)=0, | |||
:# ''μ''<sub>*</sub> is non-negative, | |||
:# If ''E'' ⊆ ''F'' then ''μ''<sub>*</sub>(''E'') ≤ ''μ''<sub>*</sub>(''F''). | |||
== Measure completion == | |||
{{main|complete measure}} | |||
Induced inner measures are often used in combination with [[outer measure]]s to extend a measure to a larger σ-algebra. If ''μ'' is a measure defined on [[σ-algebra]] ''Σ'' over ''X'' and ''μ''* and ''μ''<sub>*</sub> are corresponding induced outer and inner measures, then the sets ''T'' ∈ 2<sup>''X''</sup> such that ''μ''<sub>*</sub>(''T'') = ''μ''*(''T'') form a σ-algebra <math>\scriptstyle\hat\Sigma</math> with <math>\scriptstyle\Sigma\subseteq\hat\Sigma</math>. The set function ''μ̂'' defined by | |||
:<math>\hat\mu(T)=\mu^*(T)=\mu_*(T)</math>, | |||
for all ''T'' ∈ <math>\scriptstyle\hat\Sigma</math> is a measure on <math>\scriptstyle\hat\Sigma</math> known as the completion of ''μ''. | |||
==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) | |||
[[Category:Measures (measure theory)]] |
Latest revision as of 19:50, 4 October 2013
In mathematics, in particular in measure theory, an inner measure is a function on the set of all subsets 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 function
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).
- Superadditive: For any disjoint sets A and B,
- Infinity must be approached: If for a set A then for every positive number c, there exists a B which is a subset of A such that,
The inner measure induced by a measure
Let Σ be a σ-algebra over a set X and μ be a measure on Σ. Then the inner measure μ* induced by μ is defined by
Essentially μ* gives a lower bound of the size of any set by ensuring it is at least as big as the μ-measure of any of its Σ-measurable subsets. Even though the set function μ* is usually not a measure, μ* shares the following properties with measures:
- μ*(∅)=0,
- μ* is non-negative,
- If E ⊆ F then μ*(E) ≤ μ*(F).
Measure completion
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If μ is a measure defined on σ-algebra Σ over X and μ* and μ* are corresponding induced outer and inner measures, then the sets T ∈ 2X such that μ*(T) = μ*(T) form a σ-algebra with . The set function μ̂ defined by
for all T ∈ is a measure on known as the completion of μ.
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)