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For both a sigma algebra and a topology, we can talk about their generators.

For a topology, a base is a special generator only using union, which is a useful concept in topology. In parallel comparison, is there a similar concept for a sigma algebra like a base for a topology? How useful can it be?

Thanks and regards!

Tim
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    Technically speaking, one should impose certain restrictions on a base so that it is simple to describe. For example, every topology is a base for itself, and this does not really simplify the description of the space. Hence, are there certain conditions that you would like a base to satisfy? – Haskell Curry Jan 02 '13 at 06:45
  • Not yet. In topology, a base with the minimum cardinality will be more interesting, isn't it? – Tim Jan 02 '13 at 06:47
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    Haskell's point is well-taken, Tim. I can't see what "Not yet" means, at all. Certainly, a base of least cardinality (or any $\subseteq$-minimal base) would be more interesting than other bases, but which restrictions are *you* imposing on the "sigma algebra base" that you're looking for? Without an answer to that, we can't answer your question. – Cameron Buie Jan 02 '13 at 07:39
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    By "parallel comparison", do you mean that you are looking for a generator of a $\sigma$-algebra in a sense that by taking only unions we obtain rest of the sets? – T. Eskin Jan 02 '13 at 12:55
  • @ThomasE.: That is one possibility. A generator generating the sigma algebra by taking only (countable) union, or by something else. – Tim Jan 02 '13 at 13:51
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    Isn't this question similar to the one here: http://math.stackexchange.com/questions/77540/is-there-a-counterpart-of-a-base-of-a-topology-for-a-sigma-algebra?rq=1? – Ben West Jan 02 '13 at 20:33
  • @BDub: Good catch. I didn't remember I had asked that question... – Tim Jan 02 '13 at 21:43
  • @BDub: I think my old post asked about a particular attempt to define a base, which is then shown to be invalid by Didier. Now I am open to other possibilities to define a base of a sigma algebra. – Tim Jan 04 '13 at 01:15
  • If you are looking for an extension of some concept from one object to a similar one, then it shall at least give you some motivation in simple cases. What would you think about a basis for the Borel $\sigma$-algebra on $[0,1]$? – SBF Jan 29 '13 at 12:48
  • Algebra, monotone class, semi-ring... – André Caldas Jun 29 '13 at 01:10

1 Answers1

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A semi-field $\mathcal{S}$ on $\Omega$ is a class of subsets of $\Omega$ such that:

  • $\Omega\in\mathcal{S}$
  • $A,B\in\mathcal{S}\Rightarrow A\cap B\in\mathcal{S}$
  • $A\in\mathcal{S}\Rightarrow A^c$ can be written as a finite disjoint union of sets in $\mathcal{S}$

The class of subsets of $\Omega$ formed by taking all finite (disjoint) unions of sets in $\mathcal{S}$ form a field, $\mathcal{F}$. If we take all possible countable unions of sets in $\mathcal{F}$, then we get a $\sigma$-field.

Hence I think semifield in measure theory is an analogue of base in topology.

QED
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