User:Anghel/Sandbox

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

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

Let $A, B \in \SS$.

Then the sum


 * $\map f A + \map f B$

is well-defined in the extended real numbers $\overline \R$.

Proof
Suppose the sum $\map f A + \map f B$ is void.

By, this happens when the sum is $\paren{ +\infty } + \paren{ -\infty }$, or $\paren{ -\infty } + \paren{ +\infty }$.

, assume that $\map f A = +\infty$, and $\map f B = -\infty$.

Definition of additive function and Set Difference and Intersection form Partition give two equations: