Suppose that $\Omega = \{x_1, x_2, \dots, x_n\}$ is a finite set, and that $F = P(\Omega)$. If all outcomes are to be 'equally likely', then the probability of an event $A$ is proportional to its cardinality $n(A)$, the number of outcomes it contains. Thus, because $P(\Omega) = 1$,
$$P(A) = \frac{n(A)}{n(\Omega)} = \frac{|A|}{|\Omega|}, \quad A \subseteq \Omega$$
The probability $P$ is known as the uniform distribution on $\Omega$.
Show that the uniform distribution is a probability measure.
