Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am trying to wrap my head around random variables and can't prove the following questions:

How, in a topological space $(X, \mathcal{T})$, the collection of all Borel sets, say $\mathfrak{B}$, make a $\sigma$-algebra? And, Why is it the smallest $\sigma$-algebra containing $\mathcal{T}$?

In the introduction of the Wikipedia entry of Borel sets these two statements appear. So, I'm using the definition of Borel set given there:

A Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement.

I will appreciate any help. Thank you.

share|improve this question
1  
A Borel set is any set in a topological space that can be formed from open sets through the iterated operations of countable union, countable intersection, and relative complement. –  Michael Greinecker Nov 23 '12 at 8:01

1 Answer 1

up vote 3 down vote accepted

It makes a $\sigma$-algebra because, directly from the definition you give, it is easy to see that it is closed under taking complements and countable intersection/union. Moreover, any $\sigma$-algebra containing $\mathcal{T}$ must contain all such sets as can be formed from the open sets in $\mathcal{T}$, so the Borel sets must be the smallest $\sigma$-algebra containing $\mathcal{T}$, as all other such $\sigma$-algebras must contain it.

share|improve this answer
1  
Thanks for answering. –  Sayantan Nov 23 '12 at 6:53

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.