Additive Function on Empty Set is Zero

Theorem

Let $\AA$ be an algebra of sets.

Let $f: \AA \to \overline \R$ be an additive function on $\AA$.

Then $\map f \O = 0$.

Proof

$\O \in \AA$

Let $X \in \AA$.

Then:

 $\ds X \cap \O$ $=$ $\ds \O$ Intersection with Empty Set $\ds \leadsto \ \$ $\ds \map f X + \map f \O$ $=$ $\ds \map f {X \cup \O}$ Definition of Additive Function (Measure Theory) $\ds$ $=$ $\ds \map f X$ Union with Empty Set $\ds \leadsto \ \$ $\ds \map f \O$ $=$ $\ds 0$

$\blacksquare$