User talk:130.212.169.64

I take issue with the claim that almost sure convergence is not convergence with respect to a topology.

Sure (pointwise) convergence is convergence in the product topology on the set of real-valued functions on Omega. Almost sure (pointwise a.e.) convergence is the convergence with respect to quotient topology defined by the equivalent relation that two functions are the same if they differ on a set of measure zero. This appears to contradict the claim that almost sure convergence is not given by a topology. I do not see how that claim is justified.

Unless I am mistaken, sure convergence is equivalent to pointwise convergence. And almost sure convergence is equivalent to a.e. pointwise convergence. The difference is mostly a matter of terminology (certainly users think differently about these concepts) but logically, they are indistinguishable.

130.212.169.64 (talk) 03:09, 4 June 2019 (UTC)