Non-Negative Additive Function is Monotone

Theorem
Let $\SS$ be an algebra of sets.

Let $f: \SS \to \overline \R$ be an additive function, that is:
 * $\forall A, B \in \SS: A \cap B = \O \implies \map f {A \cup B} = \map f A + \map f B$

If $\forall A \in \SS: \map f A \ge 0$, then $f$ is monotone, that is:
 * $A \subseteq B \implies \map f A \le \map f B$

Proof
Let $A \subseteq B$.

Then: