How is this axiom useful? Today I had my first lecture of probability. The lecturer gave some rigorous definitions, and there is one part I don't understand.
Let $\Omega$ be a set, and let $\mathcal{F}$ be a set of subsets of $\Omega$. We call $\mathcal{F}$ a $\sigma$-algebra if:
1) $\Omega \in \mathcal{F}$
2) If $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$
3) For every countable sequence $(A_n)_{n \geqslant 1}$ in $\mathcal{F}$, also have $\cup_{n \geqslant 1} A_{n} \in \mathcal{F}$.
I think I can see why we'd like to have the first two axioms and how they are useful for creating the notion of an "event space", but why is the third axiom there?
 A: For one example, we'd like to be able to define the notion of a random variable that can take countably many values. Let $\Omega = \{1,2,\ldots\}$, $\mathcal F=\mathcal P(\Omega)$ (the power set of $\Omega$), and define $\mathbb P(\{n\}) = (1-p)^{n-1}p$ for some $0<p<1$. For distinct positive integers $n,m$, the sets $\{n\}$ and $\{m\}$ are disjoint, so by the countable additivity axiom,
$$
\mathbb P\left(\bigcup_{n=1}^\infty \{n\}\right) = \sum_{n=1}^\infty \mathbb P(\{n\}) = \sum_{n=1}^\infty (1-p)^{n-1}p = 1.
$$
Since $\bigcup_{n=1}^\infty \{n\} = \Omega$, it follows that $\mathbb P(\Omega)=1$ and this defines a probability measure on $(\Omega,\mathcal F)$. We can define a random variable $X:\Omega\to\mathbb R$ by $X(\omega) = \omega$ with distribution function
$$
F_X(x) = \mathbb P(\{\omega\in\Omega: X(\omega)\leqslant x\}) = \sum_{n=1}^{\lfloor x\rfloor} (1-p)^{n-1}p = 1 - (1-p)^{\lfloor x\rfloor},\ x\geqslant 0.
$$
This would not be possible if we only had finite additivity, i.e. if $A_1,\ldots,A_n\in\mathcal F$ are disjoint then $\mathbb P\left(\bigcup_{i=1}^n A_i\right) = \sum_{i=1}^n \mathbb P(A)$.
