Countably-additive

In mathematics, an additive set function is a function $\mu$ 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, $\mu(A \cup B) = \mu(A) + \mu(B).$ 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, $\mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n).$

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 $\mu$ be a set function defined on an algebra of sets $\scriptstyle\mathcal$ with values in \infty, \infty/math> (see the extended real number line). The function $\mu$ is called or , if whenever $A$ and $B$ are disjoint sets in $\scriptstyle\mathcal,$ then
$\mu(A \cup B) = \mu(A) + \mu(B).$
A consequence of this is that an additive function cannot take both $- \infty$ and $+ \infty$ as values, for the expression $\infty - \infty$ is undefined.

One can prove by mathematical induction that an additive function satisfies
$\mu\left(\bigcup_^N A_n\right)=\sum_^N \mu\left(A_n\right)$
for any $A_1, A_2, \ldots, A_N$ disjoint sets in $\mathcal.$

σ-additive set functions

Suppose that $\scriptstyle\mathcal$ is a σ-algebra. If for every sequence $A_1, A_2, \ldots, A_n, \ldots$ of pairwise disjoint sets in $\scriptstyle\mathcal,$
$\mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n),$
holds then $\mu$ 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 $\mathcal,$ we have a topology $\tau.$ If for every directed family of measurable open sets $\mathcal \subseteq \mathcal \cap \tau,$
$\mu\left(\bigcup \mathcal \right) = \sup_ \mu(G),$
we say that $\mu$ is $\tau$-additive. In particular, if $\mu$ is inner regular (with respect to compact sets) then it is $\tau$-additive.D. H. Fremlin ''Measure Theory, Volume 4'', Torres Fremlin, 2003.

Properties

Useful properties of an additive set function $\mu$ include the following.

Value of empty set

Either $\mu(\varnothing) = 0,$ or $\mu$ assigns $\infty$ to all sets in its domain, or $\mu$ assigns $- \infty$ to all sets in its domain. ''Proof'': additivity implies that for every set $A,$ $\mu(A) = \mu(A \cup \varnothing) = \mu(A) + \mu( \varnothing)$ (it's possible in the edge case of an empty domain that the only choice for $A$ is the empty set itself, but that still works). If $\mu(\varnothing) \neq 0,$ then this equality can be satisfied only by plus or minus infinity.

Monotonicity

If $\mu$ is non-negative and $A \subseteq B$ then $\mu(A) \leq \mu(B).$ That is, $\mu$ is a . Similarly, If $\mu$ is non-positive and $A \subseteq B$ then $\mu(A) \geq \mu(B).$

Modularity

A set function $\mu$ on a family of sets $\mathcal$ is called a and a if whenever $A,$ $B,$ $A\cup B,$ and $A\cap B$ are elements of $\mathcal,$ then
$\phi(A\cup B)+ \phi(A\cap B) = \phi(A) + \phi(B)$
The above property is called and the argument below proves that additivity implies modularity.

Given $A$ and $B,$ $\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B).$ ''Proof'': write $A = (A \cap B) \cup (A \setminus B)$ and $B = (A \cap B) \cup (B \setminus A)$ and $A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A),$ where all sets in the union are disjoint. Additivity implies that both sides of the equality equal $\mu(A \setminus B) + \mu(B \setminus A) + 2\mu(A \cap B).$

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 $A \subseteq B$ and $\mu(B) - \mu(A)$ is defined, then $\mu(B \setminus A) = \mu(B) - \mu(A).$

Examples

An example of a -additive function is the function $\mu$ defined over the power set of the real numbers, such that
$\mu (A)= \begin 1 & \mbox 0 \in A \\ 0 & \mbox 0 \notin A. \end$

If $A_1, A_2, \ldots, A_n, \ldots$ 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
$\mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n)$
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 $\varnothing$ to $0.$ (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 $\mu$, defined over the Lebesgue sets of the real numbers $\R$ by the formula
$\mu(A) = \lim_ \frac \cdot \lambda(A \cap (0,k)),$
where $\lambda$ denotes the Lebesgue measure and $\lim$ the Banach limit. It satisfies $0 \leq \mu(A) \leq 1$ and if $\sup A < \infty$ then $\mu(A) = 0.$

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
A_n = ,n + 1)
for $n = 0, 1, 2, \ldots$ The union of these sets is the positive reals, and $\mu$ applied to the union is then one, while $\mu$ applied to any of the individual sets is zero, so the sum of $\mu(A_n)$ is also zero, which proves the counterexample.

Generalizations

One may define additive functions with values in any additive monoid (for example any Group (mathematics), 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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* ba space – The set of bounded charges on a given sigma-algebra

References

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Measure theory
Additive functions