Sigma-fields, sigma-field generated by a collection of events

Here is a small extra comment regarding the definition of a sigma-field:

From the third axiom regarding countable unions of sets, you can deduce directly that finite unions should also be part of the sigma-field by considering a countable sequence of sets where you replace each A_m by an empty set when m > n; in this case, U_{m>=1} A_m = A_1 U ... U A_n