Show that the following intervals are element of Borel sets

The set of numbers (the real line) is $\mathbb{R}$ = {x : -∞ < x < ∞ }. Events are Borel sets of $\mathbb{R}$. To have an idea which subsets of $\mathbb{R}$ are Borel sets, do the following.

Let $I$ be the collection of all intervals of the form (a,b] = {x ϵ $\mathbb{R}$ : a < x ≤ b}, where -∞ < x ≤ b

1) Does $U_{n≥1}(0, 1- \frac{1}{n}] \in I$ ?

2) Let $B$($\mathbb{R}$) be smallest $\sigma$ - field containing $I$. Why $B$($\mathbb{R}$) is not empty?

Show that the following intervals are element of $B$($\mathbb{R}$) : {a} , [a,b] , (a,b) .

• If you show us what you have tried people at this site may be able to use that information to provide hints that use your work. – Jay Jul 10 '13 at 16:23
• Why should $\mathfrak{B}(\mathbb{R})$ be empty anyway, can a sigma-field containing a non-empty collection of subsets be ever empty? I don't get the motivation behind the second question. – Lord Soth Jul 10 '13 at 16:57