measure
Let (E,ℬ(E)) be a measurable space. A measure
on (E,ℬ(E)) is a function μ:ℬ(E)→ℝ∪{∞} with values in the extended real numbers such that:
-
1.
μ(A)≥0 for A∈ℬ(E), with equality if A=∅
-
2.
μ(⋃∞i=0Ai)=∑∞i=0μ(Ai) for any sequence of pairwise disjoint sets Ai∈ℬ(E).
Occasionally, the term positive measure is used to distinguish measures as defined here from more general notions of measure which are not necessarily restricted to the non-negative extended reals.
The second property above is called countable additivity, or σ-additivity. A finitely additive measure μ has the same definition except that ℬ(E) is only required to be an algebra and the second property above is only required to hold for finite unions. Note the slight abuse of terminology: a finitely additive measure is not necessarily a measure.
The triple (E,ℬ(E),μ) is called a measure space. If μ(E)=1, then it is called a probability space, and the measure μ is called a probability measure.
Lebesgue measure on ℝn is one important example of a measure.
Title | measure |
Canonical name | Measure |
Date of creation | 2013-03-22 11:57:33 |
Last modified on | 2013-03-22 11:57:33 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 19 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 60A10 |
Classification | msc 28A10 |
Related topic | LpSpace |
Related topic | SigmaFinite |
Related topic | Integral2 |
Related topic | Distribution |
Related topic | LebesgueMeasure |
Defines | measure space |
Defines | probability space |
Defines | probability measure |
Defines | countably additive |
Defines | finitely additive |
Defines | σ-additive |
Defines | positive measure |