Elementary Properties of Probability Measure

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 properties:

$(1): \quad \Pr \left({\varnothing}\right) = 0$

$(2): \quad \forall A \in \Sigma: \Pr \left({\complement_\Omega \left({A}\right)}\right) = 1 - \Pr \left({A}\right)$

where $\complement_\Omega \left({A}\right)$ denotes the complement of $A$ relative to $\Omega$

$(3): \quad \forall A \in \Sigma: \Pr \left({A}\right) \le 1$.

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.

$(1)$: Probability of Empty Event

From the definition of event space, we have:

• $\Omega \in \Sigma$
• $A \in \Sigma \implies \complement_\Omega \left({A}\right) \in \Sigma$

Hence as $\varnothing \cap \Omega = \varnothing$ and $\varnothing \cup \Omega = \Omega$, we have:

	\(\displaystyle \)	\(\displaystyle \Pr \left({\Omega}\right)\)	\(=\)	\(\displaystyle \Pr \left({\varnothing \cup \Omega}\right)\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \Pr \left({\varnothing}\right) + \Pr \left({\Omega}\right)\)	\(\displaystyle \)

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

$(2)$: Probability of Non-Occurence of Event

Let $A \in \Sigma$ be an event.

Then $\complement_\Omega \left({A}\right) \in \Sigma$ from the definition of event space.

From Intersection with Relative Complement, we have that $A \cap \complement_\Omega \left({A}\right) = \varnothing$.

From Union with Relative Complement, we have that $A \cup \complement_\Omega \left({A}\right) = \Omega$.

So $\Pr \left({A}\right) + \Pr \left({\complement_\Omega \left({A}\right)}\right) = 1$ from above, and so $\Pr \left({\complement_\Omega \left({A}\right)}\right) = 1 - \Pr \left({A}\right)$.

$(3)$: Probabilty Not Greater than One

From the above: $\Pr \left({A}\right) + \Pr \left({\complement_\Omega \left({A}\right)}\right) = 1$.

We have that $0 \le \Pr \left({\complement_\Omega \left({A}\right)}\right)$, hence:

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

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.3$: Example $10$