Inner measure - Alchetron, The Free Social Encyclopedia

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

φ : 2 X → [ 0 , ∞ ] ,

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,

Limits of decreasing towers: For any sequence {A_j} of sets such that A j ⊇ A j + 1 for each j and φ ( A 1 ) < ∞

Infinity must be approached: If φ ( A ) = ∞ 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

μ ∗ ( T ) = sup { μ ( S ) : S ∈ Σ and S ⊆ 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:

μ_*(∅)=0,
μ_* is non-negative,
If E ⊆ F then μ_*(E) ≤ μ_*(F).

Measure completion

Induced inner measures are often used in combination with outer measures to extend a measure to a larger σ-algebra. If μ is a finite measure defined on a 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 Σ ^ with Σ ⊆ Σ ^ . The set function μ̂ defined by

μ ^ ( T ) = μ ∗ ( T ) = μ ∗ ( T ) ,

for all T ∈ Σ ^ is a measure on Σ ^ known as the completion of μ.

References

Inner measure Wikipedia

(Text) CC BY-SA

Contents

Definition

The inner measure induced by a measure

Measure completion

References