Sigma-additive set function explained

In mathematics, an additive set function is a function $\backslash mu$ 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, $\backslash mu(A\; \backslash cup\; B)\; =\; \backslash mu(A)\; +\; \backslash mu(B).$ 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, $\backslash mu\backslash left(\backslash bigcup\_^\backslash infty\; A\_n\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash mu(A\_n).$

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 l{A} with values in [ -inf ty, inf ty] (see the extended real number line). The function \mu is called or , if whenever A and B are disjoint sets in \scriptstyle l{A}, then$$\backslash mu(A\; \backslash cup\; B)\; =\; \backslash mu(A)\; +\; \backslash mu(B).$$A consequence of this is that an additive function cannot take both - inf ty and + inf ty as values, for the expression inf ty - inf ty is undefined.

One can prove by mathematical induction that an additive function satisfies$$\backslash mu\backslash left(\backslash bigcup\_^N\; A\_n\backslash right)=\backslash sum\_^N\; \backslash mu\backslash left(A\_n\backslash right)$$for any

A _{1,} A _{2,} \ldots, A _{N} disjoint sets in $\backslash mathcal.$

σ-additive set functions

Suppose that

\scriptstyle l{A} is a σ-algebra. If for every sequence A _{1,} A _{2,} \ldots, A _{n,} \ldots of pairwise disjoint sets in \scriptstyle l{A}, $$\backslash mu\backslash left(\backslash bigcup\_^\backslash infty\; A\_n\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash mu(A\_n),$$holds then \mu is said to be or . Every -additive function is additive but not vice versa, as shown below.

τ-additive set functions

\tau. If for every directed family of measurable open sets $\backslash mathcal\; \backslash subseteq\; \backslash mathcal\; \backslash cap\; \backslash tau,$$$\backslash mu\backslash left(\backslash bigcup\; \backslash mathcal\; \backslash right)\; =\; \backslash sup\_\; \backslash mu(G),$$we say that \mu is \tau -additive. In particular, if \mu is inner regular (with respect to compact sets) then it is τ-additive.^[1]

Properties

Useful properties of an additive set function

\mu include the following.

Value of empty set

Either

\mu(\varnothing) = 0 , or \mu assigns inf ty to all sets in its domain, or \mu assigns - inf ty to all sets in its domain. Proof: additivity implies that for every set A, \mu(A) = \mu(A \cup \varnothing) = \mu(A) + \mu( \varnothing). If \mu(\varnothing) ≠ 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

See also: Valuation (geometry).

See also: Valuation (measure theory).

\mu on a family of sets l{S} is called a and a if whenever A, B, A\cup B, and A\cap B are elements of l{S}, then$$\backslash phi(A\backslash cup\; B)+\; \backslash phi(A\backslash cap\; B)\; =\; \backslash phi(A)\; +\; \backslash phi(B)$$ 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

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 $$\backslash mu\backslash left(\backslash bigcup\_^\backslash infty\; A\_n\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash mu(A\_n)$$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 . ^[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

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$$\backslash mu(A)\; =\; \backslash lim\_\; \backslash frac\; \backslash cdot\; \backslash lambda(A\; \backslash cap\; (0,k)),$$where λ denotes the Lebesgue measure and \lim the Banach limit. It satisfies 0 \leq \mu(A) \leq 1 and if \sup A < inf ty 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

\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 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

• • • • • • • • • ba space – The set of bounded charges on a given sigma-algebra

Notes and References

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