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Given the notion of some kind of convergence of sequences or nets on a set, one question is whether there exists a topology for that convergence and find it. The last section of Chapter 2 in Kelley's General Topology provides one characterization.

Since sigma algebra and topology are similar in many aspects, I was wondering if there is a concept for a sigma algebra, and/or from which people ask whether there exists a sigma algebra for a given collection of "convergent" objects and find it? Note that the concept needs not be a mimic of convergence for topology.

Although the concept may not be a mimic of convergence, I think to define something like convergence for a sigma algebra $\mathcal{F}$ on $\Omega$, one possibility is to

  • first define a "net" or "sequence" by considering a measurable mapping from a directed set $D$ or $\mathbb{N}$ with its discrete sigma algebra to the underlying set $\Omega$ of the sigma algebra $\mathcal{F}$, and

  • then define "convergence" of a "net" or "sequence" as $x \to \infty$ in $D$ or $\mathbb{N}$, and

  • then one can study what properties of the collection of all "convergent" "nets" or "sequences" has, and if they can in turn characterize the sigma algebra $\mathcal{F}$ from a given collection of "nets" or "sequences" such that the "nets" or "sequences" "converge". What do you think?

Thanks and regards!

By the way, is this kind of questions okay at MO?

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    Firstly, I don't think this question would be appropriate for MO (but of course, it is welcome here). If you get no good responses here then go ahead though. I don't have enough information to answer your question, but supposing you had a sequence $f_n$ of functions, you might want to ask if there is a measure space $(X,\Sigma,\mu)$ for which your functions are $\Sigma$-measurable, and that there exists $f$ such that $f_n \to f$ in $\mu$-measure. I don't really have any information on how you might do that or if it is an interesting question, but at least worthy of a comment.2012-12-31
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    so the way you wrote the question, I think you mean topology as the subject and not as the collection of open sets (or you would have included an article like 'a topology'). So in which case sigma algebra isn't a subject. This leads me to believe you rather ask "convergence is to a topological space as ______ is to a measure space" or the same thing as the questions with the articles included. I would say "convergence is to a topology as measurability is to a sigma algebra." This is because measure theory is a main employer of sigma algebras.2012-12-31
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    @toypajme: I should have said "convergence to a topology as what to a sigma algebra". Note that I think measurability to a sigma algebra as continuity to a topology, and to a topology, continuity and convergence are related but not the same concept.2012-12-31
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    You can use the same definition of convergence, I think, but it is not clear to me whether the theorems carry over.2013-01-01
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    @ZhenLin: Thanks! I think to define something like convergence for a sigma algebra, one possibility is to consider a measurable mapping from a directed set $D$ or $\mathbb{N}$ with its discrete sigma algebra to the underlying set of the sigma algebra, and then define "convergence" as $x \to \infty$ in $D$ or $\mathbb{N}$, and then one can study if what properties of such "convergence" has and if they can in turn characterize the sigma algebra. What do you think?2013-01-01

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