Probability of Empty Event is Zero

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

The probability measure $\Pr$ of $\mathcal E$ has the following property:
 * $\Pr \left({\varnothing}\right) = 0$

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


 * $(1): \quad \forall A \in \Sigma: 0 \le \Pr \left({A}\right)$


 * $(2): \quad \Pr \left({\Omega}\right) = 1$


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

From the definition of event space, we have:
 * $\Omega \in \Sigma$
 * $A \in \Sigma \implies \complement_\Omega \left({A}\right) \in \Sigma$

From Intersection with Empty Set:


 * $\varnothing \cap \Omega = \varnothing$

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

From Union with Empty Set:


 * $\varnothing \cup \Omega = \Omega$

Therefore we have:

As $\Pr \left({\Omega}\right) = 1$, it follows that $\Pr \left({\varnothing}\right) = 0$.