(Topology): Let $\Omega\neq \emptyset $ be an arbitrary set. A class of sets $\tau \subset 2^{\Omega} $ is
called a topology on $\Omega$ if it has the following three properties:
$(i)\,\, \emptyset,\Omega \in \tau $
$(ii)\,\, A\cap B \in \tau $ for any $A,B\in\tau$
$(iii)\,\, (\bigcup_ {A\in\mathscr {F} }A)\in \tau $ for any $\mathscr {F}\subset \tau.$
In contrast with $\sigma-$algebras, topologies are closed under finite intersections only, but
they are also closed under arbitrary unions.
(Borel $\sigma-$algebra): Let $(\Omega,\tau)$ be a topological space . The $\sigma-$algebra
$$\mathscr {B}(\Omega):= \sigma(\tau)$$
that is generated by the open sets is called the Borel $\sigma-$algebra on $\Omega$.
Consider $\Omega= (0 ,1]$. Let $\mathscr {F_{0}}$ consist of the empty
set and all sets that are finite unions of the intervals of the form
$(a,b]$. A typical element of this set is of the form$$F=(a_{1},b_{1}] \cup (a_{2},b_{2}] \cup… \cup (a_{n},b_{n}] $$
where ,$0\le a_{1}\lt b_{1}\le a_{2} \lt b_{2} \le \,… \le a_{n}
\lt b_{n}$ and $n\in \mathbb N.$Lemma:
a) $\mathscr {F_{0}}$ is an algebra
b) $\mathscr {F_{0}}$ is not a $\sigma-$algebra
c) $\sigma(\mathscr {F_{0}})=\mathscr {B}(\Omega)$
Proof (b) : "to see this ,note that $(0,\frac{n}{n+1}]\in\mathscr
{F_{0}}$ for every n , but
$\bigcup_{n=1}^{\infty}(0,\frac{n}{n+1}]=(0,1)\notin \mathscr {F_{0}} $"
Problem:
I have seen this lemma here Lecture pdf . my problem is in part (b) and (c) , if $\mathscr {F_{0}}$ is not closed under countable union , it's not closed under arbitrary unions (from $(iii)$) , and so $\mathscr {F_{0}}$ is not topology on $\Omega$ and from (Borel $\sigma-$algebra) definition I can't understand how $\sigma(\mathscr {F_{0}})=\mathscr {B}(\Omega)$ (lemma (c)). is $\mathscr {F_{0}}$ topology ? or it's not important in Borel $\sigma-$algebra? I can't fully understand what is Borel $\sigma-$algebra.Could someone help me, where I have made a mistake. Any help is greatly appreciated
Best Answer
Sometimes we call $\mathscr {F_{0}}$ as a ring. Do not try to compare it with the topology, they are different.
Actually topology is an abstraction/generalization of open sets, so for the usual topology on $\mathbb R$ they are just open sets in the normal sense, and the Borel sets are sigma algebra generated by open set - if this could help you to gain some insights.
I think the last sentence in your question actually well explained why $\mathscr {F_{0}}$ is not a sigma- algebra.
To see why $\sigma(\mathscr {F_{0}}) = \mathscr B(\Omega)$, just try to show that Borel sigma algebra is generated by each of the following collection of sets: $\{(a,b)\}$, $\{[a,b]\}$, $\{(a,b]\}$. And the other direction is obvious as every open set is in $\sigma(\mathscr {F_{0}})$.
EDIT to answer three questions in the comment