infinite product measure

Let (Ei,i,μi) be measure spacesMathworldPlanetmath, where iI an index setMathworldPlanetmathPlanetmath, possibly infiniteMathworldPlanetmathPlanetmath. We define the productPlanetmathPlanetmath of (Ei,i,μi) as follows:

  1. 1.

    let E=Ei, the Cartesian product of Ei,

  2. 2.

    let =σ((i)iI), the smallest sigma algebra containing subsets of E of the form Bi where Bi=Ei for all but a finite number of iI.

Then (E,) is a measurable spaceMathworldPlanetmathPlanetmath. The next task is to define a measure μ on (E,) so that (E,,μ) becomes in addition a measure space. Before proceeding to define μ, we make the assumptionPlanetmathPlanetmath that

each μi is a totally finite measure, that is, μi(Ei)<.

In fact, we can now turn each (Ei,i,μi) into a probability space by introducing for each iI a new measure:


With the assumption that each (Ei,i,μi) is a probability space, it can be shown that there is a unique measure μ defined on such that, for any B expressible as a product of Bii with Bi=Ei for all iI except on a finite subset J of I:


Then (E,,μ) becomes a measure space, and in particular, a probability space. μ is sometimes written μi.


  • If I is infinite, one sees that the total finiteness of μi can not be dropped. For example, if I is the set of positive integers, assume μ1(E1)< and μ2(E2)=. Then μ(B) for

    B:=B1×i>1Ei=B1×E2×i>2Ei, where B11

    would not be well-defined (on the one hand, it is μ1(B1)<, but on the other it is μ1(B1)μ2(E2)=).

  • The above construction agrees with the result when I is finite (see finite product measureMathworldPlanetmath (

Title infinite product measure
Canonical name InfiniteProductMeasure
Date of creation 2013-03-22 16:23:14
Last modified on 2013-03-22 16:23:14
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 28A35
Classification msc 60A10
Related topic ProductSigmaAlgebra
Defines totally finite measure