Talk:Product measure

Uniqueness?
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
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)

Minimal Product Measure
Hello everyone. I have some doubts that this minimal product measure mentioned in the article really exists. Take Omega to be any uncountable set. Take Sigma to be the power set of Omega. Take mu to be the measure taking any countable set to zero and any uncountable set to infinity.

Then (Omega, Sigma, mu) is a (very ugly) measure space (far from being sigma-finite).

Now I take the product of Omega with itself and consider the possible product measures on the product sigma algebra. The product sigma algebra contains all sets of the form A times B. Apart from that I dont care how the product sigma algebra looks like.

By the definition in the article every product measure satisfies mu(A times B)= mu(A) times mu (B). In particular mu(Omega times Omgea) will be infinity for any product measure.

Now, consider the maximal (Caratheodory) product measure. By construction this maximal measure takes only the values zero and infinity.

Now, lets go on to the minimal measure. The construction of the minimal measure mentioned in the article says that the minimal measure of a set is the sup over all subsets with finite maximal measure. But every set with finite measure has measure zeri. So in my example this minimal measure will be constantly zero on all measurable sets. In particular, the minimal measure of Omega times Omega will be zero. And so it is not a product measure anymore.

Is there a mistake in my argument does the existence of the minimal measure rely on some properties of the measure space which my (ugly) measure space does not have? Thank you in advance. — Preceding unsigned comment added by Fulya2000 (talk • contribs) 07:04, 8 June 2016 (UTC)

Independence
I am not very familiar with measure theory so I am not sure where would be a good place to mention it, but I think maybe the most important feature of the resulting product measure is that it models independence of random variables. I think it would be great to mention it somewhere. Clearly, the property in the definition suggests independence but for people not used to this notation it would still be nice to explicitly say it. I went and looked somewhere else (Terrence Tao's website) to make sure for example. --Technokratisch (talk) 15:56, 9 May 2020 (UTC)


 * Yes, absolutely. This article desperately needs an informal introduction and discussion before launching into formal details. 67.198.37.16 (talk) 20:40, 30 September 2020 (UTC)