complete measure

A measure spaceMathworldPlanetmath (X,𝒮,μ) is said to be completePlanetmathPlanetmathPlanetmath if every subset of a set of measure 0 is measurable (and consequently, has measure 0); i.e. if for all E𝒮 such that μ(E)=0 and for all SE we have μ(S)=0.

If a measure space is not complete, there exists a completion ( of it, which is a complete measure space (X,𝒮¯,μ¯) such that 𝒮𝒮¯ and μ¯|𝒮=μ, where 𝒮¯ is the smallest σ-algebra containing both 𝒮 and all subsets of elements of zero measure of 𝒮.

Title complete measure
Canonical name CompleteMeasure
Date of creation 2013-03-22 14:06:56
Last modified on 2013-03-22 14:06:56
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Definition
Classification msc 28A12
Related topic UniversallyMeasurable
Defines completion
Defines complete