Talk:Product measure

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Is the uniqueness of product measure guarantee only if in cases that both (X_1, \Sigma_1, \mu_1) and (X_2, \Sigma_2, \mu_2) are σ-finite? -- Jung dalglish 16:20, 8 February 2006 (UTC)

You aer right. A counterexample to the fubini theorem where one of the measures is not sigma finite is, for example, borel measure on [0, 1] times the counting measure on [0,1]. What is the measure of the diagonal set {(x,x): x \in [0, 1]}? The repeated integrals are unequal, but how do we define its measure? What do we say in this case?

Query: Is an L^1 function defined on R^2 can be approxiated with elemetary function(finite disjoint union of measurable rectangles)?

Infinite product measure[edit]

The article defined (by induction) the product of a finite number of measures. However, an antecessor article (Standard probability space) requires an infinite product measure. How it's done? In the same way that the product topology is done (sigma-algebra generated by the cylinders)? Albmont (talk) 10:23, 18 November 2008 (UTC)

Yes, basically, in the same way. However, its existence is a more delicate point. See "Probability with martingales" by David Williams, Appendix to Chapter 9. For standard measurable spaces it is easier. Note also that the product of a finite number of measures is well-defined for sigma-finite measures, while the product of a sequence of measures is well-defined for probability measures only. Boris Tsirelson (talk) 20:01, 23 March 2009 (UTC)
Another source: Section 8.2 "Infinite products of probability spaces" in the book "Real analysis and probability" by Richard M. Dudley. Boris Tsirelson (talk) 12:05, 26 March 2009 (UTC)