Real Analysis – Difference Between Borel Sigma Algebra and Cylindrical Sigma Algebra

Question

I see that there are two differen concepts for Sigma Algebras on cartesian products over the real numbers. The first one is the Borel Sigma Algebra created by the product topology. The other one is the cylindrical sigma algebra.
Actually, when I read the definition of cylinder sets via projections see Wikipedia, I first thought that these concepts would be the same. Somehow, this seems to be wrong. So how are they related to each other. Is one of them coarser than the other or are they in general incomparable and why is the cylindrical one used in probability theory?

Answer 1

The article to which you linked actually says that cylindrical $\sigma$-algebra is contained in Borel $\sigma$-algebra. The answer by Michael Greinecker shows the reverse inclusion may fail. Concerning

why would one introduce this product of sigma algebras, instead of the more natural one from topology?

I'd say that from the measure theory point of view the cylindrical $\sigma$ algebra is more natural. They both begin with the same basis of sets, finitely restricted rectangles. From here we either

1. take the smallest $\sigma$-algebra containing the rectangles, or
2. take the smallest topology containing the rectangles, and then take the smallest $\sigma$-algebra containing that topology.

Clearly, the first approach (which gives cylindrical $\sigma$-algebra) is more direct. The second approach (which gives the Boreal $\sigma$-algebra) involves taking uncountable unions of basis sets in the process of generating topology. Uncountable unions do not play well with measures.

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In "small" spaces like $\mathbb R^n$, where the $\sigma$-algebras are the same, we usually think in Borel terms. This is mostly because for analysis it is important to have close connection to the topology of Euclidean space. In huge uncountable products, there is not much that topology can do for us.