Measure of Empty Set is Zero

Theorem
Let $$\left({X, \Sigma, \mu}\right)\ $$ be a measure space.

Then $$\mu \left({\varnothing}\right) = 0\ $$.

Proof
The empty set is disjoint with itself, that is: $$\varnothing \cap \varnothing = \varnothing$$.

Hence by additivity:
 * $$\mu \left({\varnothing}\right) = \mu \left({\varnothing \cup \varnothing}\right) = \mu \left({\varnothing}\right) + \mu \left({\varnothing}\right) = 2 \mu \left({\varnothing}\right)\ $$

from which it follows directly that $$\mu \left({\varnothing}\right) = 0\ $$.