2

Why can't we describe the elements of a sigma-algebra? I was told that it is incorrect to say that a given element $B$ in a sigma-algebra $\sigma(A_i)$ can be generated by the $A_i$s using countable set operations.

Can someone provide an example of a sigma-algebra generated by some elements and a set $B$ in the sigma-algebra for which $B$ cannot be expressed in terms of the generators using countable set operations?

The Substitute
  • 3,888
  • You can describe them, of course, but not in terms of countable operations on the generating elements. Take for example the collection of open sets and look at the examples in http://math.stackexchange.com/a/73686/27978. – copper.hat Nov 07 '15 at 23:10
  • This is relevant: http://math.stackexchange.com/questions/54172/the-sigma-algebra-of-subsets-of-x-generated-by-a-set-mathcala-is-the-s/54179#54179 – Asaf Karagila Nov 07 '15 at 23:17

1 Answers1

1

I would disagree with the claim in your question, but of course it depends on what you mean by "describe." See e.g. this question by David Ullrich (and the answers), which concerns an approach to describing a Borel set via a "code" of a certain form: A construction of sigma-algebras - surely not new, right?.

SIDE NOTE: In the absence of choice, things can get weird, as we may have Borel sets without Borel codes - so in some sense, proving that Borel sets can be appropriately described requires choice. But, if we do assume choice, then the Borel sets all are describable, in a certain sense.

Community
  • 1
Noah Schweber
  • 245,398