Re: Hulls in sigma algeras

maninalift@xxxxxxxxxxxxxx wrote:


David Ulrich is correct. A countably generated or separable sigma
algebra is one generated by a countable collection of subsets.

You are both of course correct and I as usual am wrong.

But the class of sigma-algebras for which the original statement is
true is surely larger than this. I come at this from probability
theory where all of our sigma algebras are generated by random
variables or finite sets thereof. In this case we have:

E - 'microstates'
f - random variable, function E -> R
F = sigma(f) = sigma(f^-1(Borel(R)))

Here:

Intersection of all S in F containing x in E

must be f^-1({f(x})

this is clearly a member of F, is a member of F containing x and is
subsetequal to all members of F containing x.

-----

But this relies on me invoking the existence of f s.t. F=sigma(f). Is
there a more general way of classifying the sigma-algebras for which
this is true?

If you have a map of sets f;X -> Y, then a sigma algebra on Y induces
a sigma algebra on X. If the sigma algebra on Y is countably generated,
so is that on X. The same holds for a finite set of maps if you
assemble them into a map X -> Y^n. (Probably also for a countable set
of maps.) Is there a probability measure around?
--
rusty
.

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