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David Michael Roberts (Feb 28 2023 at 23:33):I'm teaching a private student some probability theory from a sophisticated point of view, and I was wondering what would be the cleanest way to construct the product measure, on the product of probability spaces. Getting the product of measurable spaces (I.e. only the data of the -algebras) is ok, but the first resources I've been looking at have been relying on a general result about extending a pre-measure like the Hahn–Kolmogorov theorem or Carathéodory's extension theorem. Is there a cleaner proof in the case of forming a product space? Obviously we can't get away from analysis completely, since we need to use completeness of the reals somewhere, I presume.
Evan Patterson (Mar 01 2023 at 04:39):In the approach to measure theory based on positive linear functionals on a function space (usually the space of continuous functions on a compact Hausdorff space), product measures are constructed by defining them on product functions and then extending to all functions by continuity. This works because the linear space spanned by product functions is dense in the space of all continuous functions. So yes, you can't get away from analysis but to my mind the approximation theorem used here has a lot more intrinsic interest than measure-theoretic nonsense like the Caratheodory extension theorem.
This approach to integration can be found in Nachbin's The Haar Integral and Vol 1 of Barry Simon's big analysis series and surely other places as well.