I know that a $\sigma$-algebra is a suitable generalization of the notion of sample space, in the following sense:
Consider a sample space $\Omega$ and a collection $\mathscr{F}$ of subsets of $\Omega$. Then $\mathscr{F}$ is called a $\sigma$-algebra on $\Omega$ if the following are satisfied: (1) $\phi \in \mathscr{F}$. (2) If $A \in \mathscr{F}$, then $A^C \in \mathscr{F}$, that is, complementation is closed. (3) If $\{A_i\}$ is a sequence in $\mathscr{F}$, then $\bigcup_i{A_i} \in \mathscr{F}$, that is, countable union is closed.
I would like to know what motivated this definition for a $\sigma$-algebra, in simple terms.