Measure (rings and algebras) question: smallest ring containing a set

[VonWeber]

Homework Statement

Let D be a class of sets of space X. There exists a unique smallest ring, R, containing D

The Attempt at a Solution

I'm working on the existence part.

If D were finite, I think that it would be possible to use set differences to find a set, D', of smaller sets from D (ones that could not be cut smaller using sets in D ).One could take all possible unions of sets in D'. I believe this would be one such set R.

However, if D were infinite or uncountable then I don't have any intuition. I'd prefer a hint rather then be given the solution...

 

[Hurkyl]

You've either omitted a lot of important information, or your question is trivially false.

e.g. given any set of 4 objects, there are 24 different rings that contain exactly those 4 objects and nothing else, organized into two isomorphism classes of 12 each. (One class is isomorphic to \mathbb{Z} / 4, and the other to \mathbb{Z} / 2 \times \mathbb{Z} / 2) And there may be more that I overlooked!

 

[VonWeber]

Rings in measure theory, as in rings and algebras. I have that in the title of my post but forgot the word 'theory'... These rings are defined to be classes of sets

closed under unions,
closed under set differences (aka relative complement),
and containing the empty set.

I think I've made a little progress on this one though. The previous problem is to show that intersections of rings (algebras, sigma-rings and sigma-algebras) are also rings (algebras, sigma-rings and sigma-algebras).

I think that is relevant. If one could show that there must be at least one ring that contains all the sets in D then the intersection of all possible rings containing D must be the smallest such set. So I guess what I'm still missing is to show that there are such rings.

 

[Hurkyl]

So I guess what I'm still missing is to show that there are such rings.

I assume earlier when you said "sets of space X" you were referring to subsets of (the set of points of) X? If so... isn't there a largest ring?

If one could show that there must be at least one ring that contains all the sets in D then the intersection of all possible rings containing D must be the smallest such set.

That sounds like a good approach.