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One can check that this function is additive by using the linearity of the limit. That this function is not σ-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.
==Generalizations==
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