Proving Borel Sets from Finite Sample Spaces

[jetoso]

Question:
To show that sets made up of single points are Borel sets, it is enough to say that:
There exist a sample space A = {a1, a2,..., an} n = 1, 2,...
then Bn = {an}; where Bn belongs to A.
Then Bn is closed, and its complement must be open.
So the sigma algebra geberated by A is a orel field because it is formed by finite unions and intersections of open sets?

I am some confused here...

 

[mathman]

In order to define Borel sets, the open sets must be defined. Once you have that, Borel sets are the smallest collection containg open sets and closed under the operations of countable unions and intersections, as well as complements.

In your example, what are the open sets?

 

[jetoso]

How about this:

Let W = (-∞, 1], be a sample space. Let the set A contained in W, be a subset of elementary events such that A = {a1, a2,...} n=1,2,..., then sigma-algebra(A) = Borel set since the complement of a single point is of the form (a, b]. If an < 1, then the complement of an is of the form (-∞, an)U(an, 1]. For an-1 < an the unions and intersections between sets also are of the form (a, b]. If an<=1 then its complement is of the form (-∞, 1), with any other union of the form (a, b] we get again a Borel set.