Standard Borel Spaces
September 19th, 2008 by WaltI 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.
September 20th, 2008 at 6:39 am
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.
September 20th, 2008 at 11:50 am
Yes.
September 22nd, 2008 at 1:04 pm
Yes it’s true, or yes this paper will answer my question?
September 22nd, 2008 at 1:34 pm
It’s true.
I think the standard reference for the topic is Parthasarathy’s Probability Measures on Metric Spaces.
November 4th, 2008 at 11:26 am
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.
November 5th, 2008 at 4:42 pm
Thanks for the Parthasarathy reference! It helped a lot!
November 11th, 2008 at 8:20 pm
You’re right. I forgot “separable”.