Measure of Empty Set is Zero

Let $$(X, \Sigma, \mu)\ $$ be a measure space. Then $$\mu(\varnothing) = 0\ $$.

Proof
$$\varnothing\cap\varnothing = \varnothing$$, so the empty set is disjoint with itself.

Hence by additivity, $$\mu(\varnothing) = \mu(\varnothing \cup \varnothing) = \mu(\varnothing) + \mu(\varnothing) = 2\mu(\varnothing)\ $$, which implies $$\mu(\varnothing) = 0\ $$.