I am studying measure theory and I have a question concerning the wikipedia-article “Product measure”. I already asked on the Wikipedia-“talk”-page but so far noone answered. The problem concerns the “minimal product measure”. I copy the following from the Wikipedia talk page:
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.