Probability of Empty Event is Zero

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

The probability measure $\Pr$ of $\EE$ has the following property:
 * $\map \Pr \O = 0$

Proof
From the conditions for $\Pr$ to be a probability measure, we have:


 * $(1): \quad \forall A \in \Sigma: 0 \le \map \Pr A$


 * $(2): \quad \map \Pr \Omega = 1$


 * $(3): \quad \displaystyle \map \Pr {\bigcup_{i \mathop \ge 1} A_i} = \sum_{i \mathop \ge 1} \map \Pr {A_i}$ where all $A_i$ are pairwise disjoint.

From the definition of event space, we have:
 * $\Omega \in \Sigma$
 * $A \in \Sigma \implies \relcomp \Omega A \in \Sigma$

From Intersection with Empty Set:


 * $\O \cap \Omega = \O$

Therefore $\O$ and $\Omega$ are pairwise disjoint.

From Union with Empty Set:


 * $\O \cup \Omega = \Omega$

Therefore we have:

As $\map \Pr \Omega = 1$, it follows that $\map \Pr \O = 0$.

Also see

 * Elementary Properties of Probability Measure