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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?

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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.

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  • $\begingroup$ By "describe" I mean write $B$ in terms of the generators of the sigma-algebra using countable set operations. $\endgroup$ – The Substitute Nov 8 '15 at 10:02

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