(1). According to Wikipedia, a field of subsets of $X$ is defined to be a non-empty subset of the power set of $X$ closed under the intersection and union of pairs of sets and under complements of individual sets. I was wondering if there is any redundancy in this definition?

Can it be defined to be closed only under pairwise intersection and pairwise union? Or only closed under complement and pairwise intersection (or pairwise union)? If yes in each case, how can we show it also closed under the other operation?

(2). A sigma algebra of subsets of $X$ is defined to be a nonempty subset of the power set of $X$ closed under complementation and under countable union.

Can it be defined to be closed under countable union and countable intersection? If yes, how can we show it is subsequently closed under complement?

Thanks and regards!

  • $\begingroup$ Please try to use existing tags, rather than invent new ones. You tagged this with a brand-new tag "set", when there was a perfectly fine "set-theory" tag alread in existence; same with "field" vs "field-theory". $\endgroup$ – Arturo Magidin Oct 12 '10 at 3:19
  • $\begingroup$ @Arturo: Okay! I see. $\endgroup$ – Tim Oct 12 '10 at 3:25

No, you can't get complements from unions and intersections. For example, let $X$ be a nonempty set. Then $\{\emptyset\}$ is nonempty, closed under (arbitrary) intersection and union, but not closed under complements.

You can get intersections from unions and complements using De Morgan's laws. To get an intersection, just take the complement of the union of the complements. Similarly, you get unions from intersections and complements. So yes, the definition of field of subsets was redundant.

  • $\begingroup$ In en.wikipedia.org/wiki/…, it says "If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebra." Does it mean sigma algebra is defined to be closed only under countable intersection and countable union? Or does "an algebra over a set" already mean that it is also defined to be closed under complement? $\endgroup$ – Tim Oct 11 '10 at 22:12
  • $\begingroup$ The latter; "algebra of sets" and "field of sets" are synonyms, both being nonempty, closed under complements, and closed under finite unions (and hence closed under finite intersections). The authors of that article were just inconsistent in their terminology, and the different terminology should be mentioned somewhere on that page but isn't. See also eom.springer.de/A/a011400.htm $\endgroup$ – Jonas Meyer Oct 11 '10 at 22:15

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