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

From Properties of Algebras of Sets:

$\O \in \AA$

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

\(\ds \leadsto \ \ \)

\(\ds \map f \O\)

\(=\)

\(\ds 0\)

$\blacksquare$