σ-finite
A measure space (Ω,ℬ,μ) is a finite measure space if μ(Ω)<∞; it is σ-finite if the total space is the union of a finite or countable
family of sets of finite measure, i.e. if there exists a countable set ℱ⊂ℬ such that μ(A)<∞ for each A∈ℱ, and
Ω=⋃A∈ℱA.
In this case we also say that μ is a σ-finite measure.
If μ is not σ-finite, we say that it is σ-infinite
.
Examples. Any finite measure space is σ-finite. A more interesting example is the Lebesgue measure μ in ℝn: it is σ-finite but not finite. In fact
ℝn=⋃k∈ℕ[-k,k]n |
([-k,k]n is a cube with center at 0 and side length 2k, and its measure is (2k)n), but μ(ℝn)=∞.
Title | σ-finite |
---|---|
Canonical name | sigmafinite |
Date of creation | 2013-03-22 12:29:48 |
Last modified on | 2013-03-22 12:29:48 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 13 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 28A10 |
Synonym | σ finite |
Synonym | sigma-finite |
Synonym | sigma finite |
Related topic | Measure |
Related topic | MeasureSpace |
Related topic | AlternativeDefinitionOfSigmaFiniteMeasure |
Related topic | AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure |
Defines | σ-infinite |
Defines | sigma-infinite |
Defines | finite measure space |