Probability 1 -- Sigma Algebra

Definition in this paper mainly comes from Probability and Statistical Inference, Second Edition, Robert Bartoszynski, Magdalena Niewiadomska-bugaj

Infinite Operations on Events

There are correspondences between infinite union, intersection and normally said words. They are listed as below:

“at least one \(A_i\) occurs": \(\overset{\infty}{\underset{i = 1}{\cup}} A_i\)

"all \(A_i\) occurs": \(\overset{\infty}{\underset{i = 1}{\cap}} A_i\)

"infinitely many \(A_i\) occurs": \(\lim \sup A_n := \overset{\infty}{\underset{i = 1}{\cap}} \overset{\infty}{\underset{k = i}{\cup}} A_k\)

"all except finitely many \(A_i\) occurs": \(\lim \inf A_n := \overset{\infty}{\underset{i = 1}{\cup}} \overset{\infty}{\underset{k = i}{\cap}} A_k\)

Then it comes the definition of limit of infinite sequence of set:

Definition: Limit

If the sequence \(A_1, A_2, ...\) is increasing, then \[\lim A_n = \overset{\infty}{\underset{i = 1}{\cup}} A_i.\]If the sequence \(A_1, A_2, ...\) is decreasing, then\[ \lim A_n = \overset{\infty}{\underset{i = 1}{\cap}} A_i.\]

Definition: \(\sigma\)-algebra

A non-empty class \(\mathcal{A}\) of subsets of \(\mathcal{S}\) that is closed under complementation, and all countable operations(union, intersection) is called a \(\sigma\)-algebra.

The \(\sigma\)-algebra can be generated by first selecting some elements then do operations on them. Therefore we have the following theorem.

Theorem: Generated \(\sigma\)-algebra

For any nonempty class \(\mathcal{K}\) of subsets of \(\mathcal{S}\), there exists a smallest \(\sigma\)-algebra that contains all sets in \(\mathcal{K}.\) It is then called the \(\sigma\)-algebra generated by \(\mathcal{K}.\)

cite:

Probability and Statistical Inference, Second Edition, Robert Bartoszynski, Magdalena Niewiadomska-bugaj