Sigma-additive set function
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, formula_1 If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of "k" disjoint sets (where "k" is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the  terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an "infinite" number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, formula_2 Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity. The term modular set function is equivalent to additive set  function; see modularity below. Additive (or finitely additive) set functions. Let formula_3 be a set function defined on an algebra of sets formula_4 with values in formula_5 (see the extended real number line). The function formula_3 is called additive, or finitely additive, if, whenever formula_7 and formula_8 are disjoint sets in formula_9 one has formula_10 A consequence of this is that an additive function cannot take both formula_11 and formula_12 as values, for the expression formula_13 is undefined. One can prove by mathematical induction that an additive function satisfies formula_14 for any formula_15 disjoint sets in formula_16 σ-additive  set functions. Suppose that formula_4 is a σ-algebra. If for every sequence formula_18 of pairwise disjoint sets in formula_9 formula_20 holds then formula_3 is said to be or . Every -additive function is additive but not vice versa, as shown below. τ-additive set functions. Suppose that in addition to a sigma algebra formula_22 we have a topology formula_23 If for every directed family of measurable open sets formula_24 formula_25 we say that formula_3 is formula_27-additive. In particular, if formula_3 is inner regular (with respect to compact sets) then it is τ-additive. Properties. Useful properties of an additive  set function formula_3 include the following. Value of empty set. Either formula_30 or formula_3 assigns formula_32 to all sets in its domain, or formula_3 assigns formula_11 to all sets in its domain. "Proof": additivity implies that for every set formula_35 formula_36 If formula_37 then this equality can be satisfied only by plus or minus infinity. Monotonicity. If formula_3 is non-negative and formula_39 then formula_40 That is, formula_3 is a "monotone set function". Similarly, If formula_3 is non-positive and formula_39 then formula_44 Modularity. Given formula_7 and formula_46 formula_47 "Proof": write formula_48 and formula_49 and formula_50 where all sets in  the union are disjoint. Additivity implies that both sides of the equality equal formula_51 The above property is called "modularity", and we have just proved that modularity is equivalent to additivity. However, there are related properties called "submodularity" and "subadditivity", which are not equivalent. Note that modularity has a different and unrelated meaning in the context of complex functions; see modular form. Set difference. If formula_39 and formula_53 is defined, then formula_54 Examples. An example of a -additive function is the function formula_3 defined over the power set of the real numbers, such that formula_56 If formula_18 is a sequence  of disjoint sets of real numbers, then either none of the sets contains 0, or precisely one of them does. In either case, the equality formula_58 holds. See measure and signed measure for more examples of -additive functions. An additive function which is not σ-additive. An example of an additive function which is not σ-additive is obtained by considering formula_3, defined over the Lebesgue sets of the real numbers formula_60 by the formula formula_61 where formula_62 denotes the Lebesgue measure and formula_63 the Banach limit. It satisfies formula_64 and if formula_65 then formula_66 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 formula_67 for formula_68 The union of these sets is the positive reals, and formula_3 applied to the union is then one, while formula_3 applied to any of the individual sets is zero, so the sum of formula_71is also zero, which proves the counterexample. Generalizations. One may define additive functions with values in any additive monoid (for example any group or more commonly a vector space). For sigma-additivity, one needs in addition that the concept  of limit of a sequence be defined on that set. For example, spectral measures are sigma-additive functions with values in a Banach algebra. Another example, also from quantum mechanics, is the positive operator-valued measure.