measurable space

A measurable spaceMathworldPlanetmathPlanetmath is a set E together with a collectionMathworldPlanetmath of subsets of E which is a sigma algebra.

The elements of are called measurable sets.

A measurable space is the correct object on which to define a measureMathworldPlanetmath; will be the collection of sets which actually have a measure. We normally want to ensure that contains all the sets we will ever want to use. We usually cannot take to be the collection of all subsets of E because the axiom of choiceMathworldPlanetmath often allows one to construct sets that would lead to a contradictionMathworldPlanetmathPlanetmath if we gave them a measure (even zero). For the real numbers, Vitali’s theorem states that cannot be the collection of all subsets if we hope to have a measure that returns the length of an open interval.

Title measurable space
Canonical name MeasurableSpace
Date of creation 2013-03-22 11:57:30
Last modified on 2013-03-22 11:57:30
Owner djao (24)
Last modified by djao (24)
Numerical id 11
Author djao (24)
Entry type Definition
Classification msc 28A33
Defines measurable set