Editing Sigma-additive set function (section)

==Additive (or finitely additive) set functions==

Let <math>\mu</math> be a [[set function]] defined on an [[Field of sets|algebra of sets]] <math>\scriptstyle\mathcal{A}</math> with values in <math>[-\infty, \infty]</math> (see the [[extended real number line]]). The function <math>\mu</math> is called '''{{visible anchor|additive|additive set function}}''' or '''{{visible anchor|finitely additive|finitely additive set function}}''', if whenever <math>A</math> and <math>B</math> are [[disjoint set]]s in <math>\scriptstyle\mathcal{A},</math> then
<math display=block>\mu(A \cup B) = \mu(A) + \mu(B).</math>
A consequence of this is that an additive function cannot take both <math>- \infty</math> and <math>+ \infty</math> as values, for the expression <math>\infty - \infty</math> is undefined.

One can prove by [[mathematical induction]] that an additive function satisfies
<math display=block>\mu\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu\left(A_n\right)</math>
for any <math>A_1, A_2, \ldots, A_N</math> disjoint sets in <math display=inline>\mathcal{A}.</math>