# Sigma algebra generated by a particular collection.

Let $F$ be the collection of all subsets of $\mathbb{R}$ with exactly three elements. Describe the $\sigma$-algebra of $\mathbb{R}$ generated by $F$. One thing is clear about collection $F$ which has infinite members. To get $\sigma$-algebra we include the sets formed by taking countable unions and intersections of members of $F$ and we do include the complements. Now I found it is going to be very large sigma algebra but how can I compare it with Borel $\sigma$-algebra of $\mathbb{R}$? I am looking for your valuable hints.

• Hints: Show that every finite set is in F, that the collection of finite sets generates the same sigma-algebra as F, and determine the sigma-algebra generated by all the finite sets.
– Did
Commented Sep 29, 2015 at 17:19
• @Did Why not put that in an answer?
– user231101
Commented Sep 29, 2015 at 17:46
• thank you but then finally what is the sigma algebra look likes? Commented Sep 29, 2015 at 17:47
• @MikeHaskel Because my hope is that the OP will transform these succinct (but precise, or so I think) hints into a full proof and, doing so, will have truly benefitted from their contact with the site (as opposed to, say, receiving a fully written answer ready to be handed to their TA). Call me an idealist all you want...
– Did
Commented Sep 29, 2015 at 17:49
• @Did I meant, why not literally put the hint you wrote in the answer box. I would up-vote it.
– user231101
Commented Sep 29, 2015 at 17:54

1. Show that you can write any three-element set as a finite combination (via union, complement, intersection) of singletons, and vice versa. What does that tell you about the $\sigma$-algebras these families (singletons and three-element sets) generate?
2. Three-element sets are closed, and therefore Borel. Keeping in mind that the $\sigma$-algebra generated by a family is the smallest $\sigma$-algebra containing that family, what does that tell you about the relationship between your $\sigma$-algebra and the Borel $\sigma$-algebra?