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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!

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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 2 '13 at 6:45
    
Not yet. In topology, a base with the minimum cardinality will be more interesting, isn't it? –  Tim Jan 2 '13 at 6:47
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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 2 '13 at 7:39
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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? –  Thomas E. Jan 2 '13 at 12:55
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Isn't this question similar to the one here: math.stackexchange.com/questions/77540/…? –  Ben Jan 2 '13 at 20:33

1 Answer 1

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.

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