""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if:
(A.1) $\Omega\in\mathcal F$
(A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$
(A.3) $\ A_1,A_2,…\in\mathcal F\implies\bigcup _{n=1}^\infty A_n\in\mathcal F$
To make things work, need a bit more: namely, that we are allowed to take countable unions.""
I am struggling to understand what (A.3) is trying to say. Does $\bigcup _{n=1}^\infty A_n$ mean a union across all $\ A_n$? In addition, what is the significance of (A.3)? Are $\ A_n$ subsets of A? What are countable sets?
Best Answer
Each $A_n$ is a subset of $\Omega$. In the previous axiom we only needed to refer to one arbitrary set, so we just called it $A$. Now we want a collection of countably many arbitrary subsets $A_1,A_2, A_3,A_4,A_5,...$ so we need to label them somehow, and we do that via subscripts running over the natural numbers.
Now remember, a $\sigma$-algebra is a collection of subsets of $\Omega$ satisfying some rules. The rule that (A.3) captures is that if you take countably many subsets of $\Omega$ each of which are in $\mathcal{F}$ then the union (which is another subset of $\Omega$), is still a set in $\mathcal{F}$.
Does that help?