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A [[set function]] <math>\mu</math> on a [[family of sets]] <math>\mathcal{S}</math> is called a '''{{visible anchor|modular set function}}''' and a '''[[Valuation (geometry)|{{visible anchor|valuation}}]]''' if whenever <math>A,</math> <math>B,</math> <math>A\cup B,</math> and <math>A\cap B</math> are elements of <math>\mathcal{S},</math> then
<math display="block"> \
The above property is called '''{{visible anchor|modularity}}''' and the argument below proves that additivity implies modularity.
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See [[Measure (mathematics)|measure]] and [[signed measure]] for more examples of {{sigma}}-additive functions.
A ''charge'' is defined to be a finitely additive set function that maps <math>\varnothing</math> to <math>0.</math><ref>{{Cite book|
===An additive function which is not σ-additive===
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