# Elementary Properties of Probability Measure

## Theorem

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

The probability measure $\Pr$ of $\EE$ has the following properties:

### Probability of Empty Event is Zero

$\map \Pr \O = 0$

### Probability of Event not Occurring

$\forall A \in \Sigma: \map \Pr {\Omega \setminus A} = 1 - \map \Pr A$

### Probability Measure is Monotone

Let $A, B \in \Sigma$ such that $A \subseteq B$.

Then:

$\map \Pr A \le \map \Pr B$