I just spotted this article on arxiv: Some Notes on Standard Borel and Related Spaces. A standard Borel space is a set with a σ-algebra which can be realized as the set of Borel sets of a complete metric space. The paper is an attempt to describe the theory of standard Borel spaces with the minimum of reliance on metric or topological ideas.
Yay! Not too late for the paper I’m writing with Derek Wise, Aristide Baratin and Laurent Freidel… the one that made us learn about Polish spaces.
If you take a Polish space and think of it as a set with a sigma-algebra of subsets, is this the same as a standard Borel space? I thought I convinced myself that this was true.
Maybe this paper will answer that question.
Yes.
Yes it’s true, or yes this paper will answer my question?
It’s true.
I think the standard reference for the topic is Parthasarathy’s Probability Measures on Metric Spaces.
By the way, I think you gave the wrong definition of ’standard Borel space’. It’s a set with a σ-algebra which can be realized as the set of Borel sets of a separable complete metric space.
I think this extra word eliminates some pathologically large examples.
Thanks for the Parthasarathy reference! It helped a lot!
You’re right. I forgot “separable”.
cn a borel metric be called a topology: BOREL TOPOLOGY?