Editing Sigma-additive set function (section)

===An additive function which is not &sigma;-additive===

An example of an additive function which is not &sigma;-additive is obtained by considering <math>\mu</math>, defined over the Lebesgue sets of the [[real number]]s <math>\R</math> by the formula
<math display=block>\mu(A) = \lim_{k\to\infty} \frac{1}{k} \cdot \lambda(A \cap (0,k)),</math>
where <math>\lambda</math> denotes the [[Lebesgue measure]] and <math>\lim</math> the [[Banach limit]]. It satisfies <math>0 \leq \mu(A) \leq 1</math> and if <math>\sup A < \infty</math> then <math>\mu(A) = 0.</math>

One can check that this function is additive by using the linearity of the limit. That this function is not &sigma;-additive follows by considering the sequence of disjoint sets
<math display=block>A_n = [n,n + 1)</math>
for <math>n = 0, 1, 2, \ldots</math> The union of these sets is the [[positive reals]], and <math>\mu</math> applied to the union is then one, while <math>\mu</math> applied to any of the individual sets is zero, so the sum of <math>\mu(A_n)</math> is also zero, which proves the counterexample.