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, 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 equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function. 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 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 be a set function defined on an algebra of sets with values in . The function is called ' or ', 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

Suppose that is a σ-algebra. If for every sequence of pairwise disjoint sets in
holds then 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 we have a topology If for every directed family of measurable open sets
we say that is -additive. In particular, if is inner regular then it is -additive.

Properties

Useful properties of an additive set function include the following.

Value of empty set

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

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 Valuation | if whenever and are elements of then
The above property is called ' 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

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

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 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.