infinity. Monotonicity If is non-negative and then That is, is a . Similarly, If is non-positive and then Modularity A set function on a family of sets is called a and a if whenever and are elements of then The above property is called and the argument below proves that modularity is equivalent to additivity. Given and Proof: write and and where all sets in the union are disjoint. Additivity implies that both sides of the equality equal However, the related properties of submodularity and subadditivity are not equivalent to each other. Note that modularity has a different and unrelated meaning in the context of complex functions; see modular form. Set difference If and is defined, then Examples An example of a -additive function is the function defined over the power set of the real numbers, such that If 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 holds. See measure and signed measure for more examples of -additive functions. A charge is defined to be a finitely additive set function that maps to (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range is a bounded subset of R.) An additive function which is not σ-additive An example of an additive function which is not σ-additive is obtained by considering , defined over the Lebesgue sets of the real numbers by the formula where denotes the Lebesgue measure and the Banach limit. It satisfies and if then 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 for The union of these sets is the positive reals, and applied to the union is then one, while applied to any of the individual sets is zero, so the sum of is 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. See