Probability Measure is Monotone/Proof 2

Theorem
Let $\mathcal E$ be an experiment with probability space $\left({\Omega, \Sigma, \Pr}\right)$.

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

Then:


 * $\Pr \left({A}\right) \le \Pr \left({B}\right)$

where $\Pr \left({A}\right)$ denotes the probability of event $A$ occurring.

Proof
As by definition a probability measure is a measure, we can directly use the result Measure is Monotone.

Also see

 * Elementary Properties of Probability Measure