The question is self-describing.
Is there notation denoting that one sigma-algebra is sub-sigma-algebra of another?
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measure-theory
notation
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0I think this is reasonable: http://www.proofwiki.org/wiki/Definition:Sub-Sigma-Algebra. – 2012-08-06
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0@unit3000-21 the same as for sets? – 2012-08-06
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0because they are essentially sets – 2012-08-06
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0I believe so. This seems to be confirmed in two other books I have. But don't try to generalize "because they are essentially sets," since for groups, you will see $H \leq G$ or but (probably) not $H \subset G$. – 2012-08-06
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0@unit3000-21 By definition it is a collection of subsets of a set stable under finitely many set operations + contains all infinite unions of subsets from it. – 2012-08-06
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0@unit3000-21 if you put what you wrote as an answer - I will accept it. Note: later I took a look at filtration for martingales - and this notation was used as from definition. – 2012-08-08
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0As much as I'd love a +15, the content of Nate's answer below is essentially what I wrote in the comment, so accepting his would serve the same purpose. – 2012-08-13
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0@unit3000-21 yes and no. but i did as you asked. – 2012-08-13
1 Answers
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I have never seen such a notation, per se. The closest I've seen is something like:
Let $\mathcal{F}, \mathcal{G}$ be $\sigma$-algebras, with $\mathcal{F} \subset \mathcal{G}$.
That is, using $\subset$ (to indicate containment as sets), where it is made clear elsewhere (or from context) that the sets in question are $\sigma$-algebras.