Lerche–Newberger sum rule

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

$\phi :2^X\to [0,\mathrm{\infty }],$

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).

$\phi (\varnothing )=0$

• Superadditive: For any disjoint sets A and B,

$\phi (A\cup B)\ge \phi (A)+\phi (B).$

• Limits of decreasing towers: For any sequence {A_j} of sets such that $A_j\supseteq A_{j+1}$ for each j and $\phi (A_1)<\mathrm{\infty }$

$\phi \left(\bigcap _{j=1}^{\mathrm{\infty }}{A}_j\right)=\underset{j\to \mathrm{\infty }}{lim}\phi (A_j)$

• Infinity must be approached: If $\phi (A)=\mathrm{\infty }$ for a set A then for every positive number c, there exists a B which is a subset of A such that,

$c\le \phi (B)<\mathrm{\infty }$

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

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

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:

1. μ_*(∅)=0,
2. μ_* is non-negative,
3. 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 ∈ 2^X such that μ_*(T) = μ*(T) form a σ-algebra $\hat{\Sigma }$ with $\Sigma \subseteq \hat{\Sigma }$. The set function μ̂ defined by

$\hat{\mu }(T)={\mu }^{\ast }(T)={\mu }_{\ast }(T)$,

for all T ∈ $\hat{\Sigma }$ is a measure on $\hat{\Sigma }$ 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)