Sigma-additive set function on arvau

Wikipedia / 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 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 -additive set function is a function that has the additivity property even for countably infinite

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 -additivity; that is, -additivity implies additivity.

The term modular set function is equivalent to additive set function; see modularity below.

Additive (or finitely additive) set functions

be a set function defined on an algebra of sets (see the extended real number line is called or are disjoint sets is undefined.

One can prove by mathematical induction

σ-additive set functions

is a σ-algebra. If for every sequence is said to be or . Every -additive function is additive but not vice versa, as shown below.

τ-additive set functions

we have a topology If for every directed family of measurable open sets is inner regular-additive.

Properties

include the following.

Value of empty set

to all sets in its domain. Proof is the empty set then this equality can be satisfied only by plus or minus infinity.

Monotonicity

is a

Modularitymodularity

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

Proof

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

Examples

An example of a defined over the power set of the real numbers

holds.

See measure and signed measure for more examples of -additive functions.

A charge (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 , defined over the Lebesgue sets of the real numbers denotes the Lebesgue measure the Banach limit

One can check that this function is additive by using the linearity of the limit. That this function is not The union of these sets is the positive reals 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.