Additive Function on Empty Set is Zero

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Let $\AA$ be an algebra of sets.

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

Then $\map f \O = 0$.


From Properties of Algebras of Sets:

$\O \in \AA$

Let $X \in \AA$.


\(\ds X \cap \O\) \(=\) \(\ds \O\) 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
\(\ds \leadsto \ \ \) \(\ds \map f \O\) \(=\) \(\ds 0\)