product σ-algebra

Given measurable spacesMathworldPlanetmathPlanetmath (E,) and (F,𝒢), the productPlanetmathPlanetmathPlanetmath σ-algebra ×𝒢 is defined to be the σ-algebra on the Cartesian product E×F generated by sets of the form A×B for A and B𝒢.


More generally, the product σ-algebra can be defined for an arbitrary number of measurable spaces (Ei,i), where i runs over an index setMathworldPlanetmathPlanetmath I. The product ii is the σ-algebra on the generalized cartesian product iEi generated by sets of the form iAi where Aii for all i, and Ai=Ei for all but finitely many i. If πj:iEiEj are the projection maps, then this is the smallest σ-algebra with respect to which each πj is measurable (

Title product σ-algebra
Canonical name Productsigmaalgebra
Date of creation 2013-03-22 18:47:21
Last modified on 2013-03-22 18:47:21
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Definition
Classification msc 28A60
Synonym product sigma-algebra
Related topic ProductMeasure
Related topic InfiniteProductMeasure