Addition Law of Probability/Proof 1

Theorem
Let $\Pr$ be a probability measure on an event space $\Sigma$.

Let $A, B \in \Sigma$.

Then:
 * $\Pr \left({A \cup B}\right) = \Pr \left({A}\right) + \Pr \left({B}\right) - \Pr \left({A \cap B}\right)$

That is, the probability of either event occurring equals the sum of their individual probabilities less the probability of them both occurring.

Proof
By definition, a probability measure is a measure.

Hence, again by definition, it is a countably additive function.

By Measure is Finitely Additive Function, we have that $\Pr$ is an additive function.

So Additive Function on Union of Sets can be applied directly.