In Probability,Random Variables book by Athanasios Papoulis, this is given
Borel fields. Suppose that $A_1$,...$A_n$,.... $\;$ is an infinite sequence of sets in $F$.If the union and intersection of these sets also belongs to $F$,then $F$ is called a Borel Field.
The class of all subsets of a set $S$ (the sample space) is a Borel field. Suppose that $C$ is a class of subsets of $S$ that is not a field. It can be shown that there exists a smallest Borel field containing all the elements of $C$.
Example:
Suppose that $S$ consists of the four elements a,b,c,d and $C$ consists of the sets {a} and {b}. Attaching to $C$ the complements of {a} and {b} and their unions and intersections, we conclude that the smallest field containing {a} and {b} consists of the sets
{$\emptyset$}$\quad${a}$\quad${b}$\quad${a,b}$\quad${c,d}$\quad${b,c,d}$\quad${a,c,d}$\quad$$S$
Events. In probability theory, events are certain subsets of $S$ forming a Borel field.
I do understand what a field is, but i am unable to understand the above paragraph on Borel fields and its connection to probability theory. I am not too familiar with set theory, please help me understand what the above paragraph means.
Thanks in advance
So probability is defined as a measure $\mathbb P$ over some topological space $\Omega$.
The measure $\mathbb P$ is a map that maps subsets (events) of $\Omega$ into a real number that is between $0$ and $1$. But it needs to meet some criteria, basically we want: $$\mathbb P(\emptyset) =0$$ $$\mathbb P(\Omega) =1$$ For events pairwise disjoint $\{A_i\}_{i =1}^{\infty}$ $$\mathbb P(\bigcup_{i =1}^{\infty}A_i)=\sum_{I=1}^{\infty}\mathbb P(A_i)$$
In order for the above to be true, we found that usually we cannot define a proper $\mathbb P$ for all subsets of $\Omega$, but only for subsets consisting a $\sigma$-algebra.
And for probability it's best to define it over the $\sigma$-algebra generated by open sets of the topological space $\Omega$. For example, it will be most convenient to keep the measurability for composite of random variables.