Sigma-additive set function

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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, 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,

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

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Let   be a set function defined on an algebra of sets   with values in   (see the extended real number line). The function   is called additive or finitely additive, if whenever   and   are disjoint sets in   then   A consequence of this is that an additive function cannot take both   and   as values, for the expression   is undefined.

One can prove by mathematical induction that an additive function satisfies   for any   disjoint sets in  

σ-additive set functions

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Suppose that   is a σ-algebra. If for every sequence   of pairwise disjoint sets in     holds then   is said to be countably additive or 𝜎-additive. Every 𝜎-additive function is additive but not vice versa, as shown below.

τ-additive set functions

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Suppose that in addition to a sigma algebra   we have a topology   If for every directed family of measurable open sets     we say that   is  -additive. In particular, if   is inner regular (with respect to compact sets) then it is τ-additive.[1]

Properties

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Useful properties of an additive set function   include the following.

Value of empty set

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Either   or   assigns   to all sets in its domain, or   assigns   to all sets in its domain. Proof: additivity implies that for every set     If   then this equality can be satisfied only by plus or minus infinity.

Monotonicity

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If   is non-negative and   then   That is,   is a monotone set function. Similarly, If   is non-positive and   then  

Modularity

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A set function   on a family of sets   is called a modular set function and a valuation if whenever       and   are elements of   then   The above property is called modularity and the argument below proves that additivity implies modularity.

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

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If   and   is defined, then  

Examples

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An example of a 𝜎-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 𝜎-additive functions.

A charge is defined to be a finitely additive set function that maps   to  [2] (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

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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

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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 also

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This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

References

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  1. ^ D. H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.
  2. ^ Bhaskara Rao, K. P. S.; Bhaskara Rao, M. (1983). Theory of charges: a study of finitely additive measures. London: Academic Press. p. 35. ISBN 0-12-095780-9. OCLC 21196971.