I need to show that these generated $\sigma$-algebras are the same:
$F_1 = \sigma(\{[a,b) : -\infty
$F_2 = \sigma(\{(-\infty , x] :x \in \mathbb{R} \})$
I am not sure, but my idea thus far:
- Show that every interval $[a,b)$ is an element of $\{(-\infty , x) :x \in \mathbb{R} \}$.
- Show that every interval $(-\infty , x]$ is an element of $\{[a,b) : -\infty.
If my idea is right (please tell me if I'm wrong), I am having trouble trying to express some things. For example, I want to show that I can choose an $a$ and a $b$ such that I get the open interval $(x,\infty)$, a complement of an element expressed in $\{(-\infty , x) :x \in \mathbb{R} \}$. Does it make sense to say something like $a=x+\epsilon : \epsilon \rightarrow 0$ and $lim_{n\rightarrow{}\infty}b_n = \infty$? I am not sure if I am expressing that $a$ approaches a number $x$, and that $b$ approaches $\infty$!
Thanks!