Sigma additivity

In mathematics, additivity (specifically finite additivity) and sigma additivity (also called countable additivity) of a function (often a measure) defined on subsets of a given set 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.

Additive (or finitely additive) set functions

Let [math]\displaystyle{ \mu }[/math] be a function defined on an algebra of sets [math]\displaystyle{ \scriptstyle\mathcal{A} }[/math] with values in [−∞, +∞] (see the extended real number line). The function [math]\displaystyle{ \mu }[/math] is called additive, or finitely additive, if, whenever A and B are disjoint sets in [math]\displaystyle{ \scriptstyle\mathcal{A} }[/math], one has

[math]\displaystyle{ \mu(A \cup B) = \mu(A) + \mu(B). \, }[/math]

(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

[math]\displaystyle{ \mu\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu(A_n) }[/math]

for any [math]\displaystyle{ A_1,A_2,\dots,A_N }[/math] disjoint sets in [math]\displaystyle{ \scriptstyle\mathcal{A} }[/math].

σ-additive set functions

Suppose that [math]\displaystyle{ \scriptstyle\mathcal{A} }[/math] is a σ-algebra. If for any sequence [math]\displaystyle{ A_1,A_2,\dots,A_n,\dots }[/math] of pairwise disjoint sets in [math]\displaystyle{ \scriptstyle\mathcal{A} }[/math], one has

[math]\displaystyle{ \mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n) }[/math],

we say that μ is countably additive or σ-additive.
Any σ-additive function is additive but not vice versa, as shown below.

τ-additive set functions

Suppose that in addition to a sigma algebra [math]\displaystyle{ \scriptstyle\mathcal{A} }[/math], we have a topology τ. If for any directed family of measurable open sets [math]\displaystyle{ \scriptstyle\mathcal{G} }[/math] ⊆ [math]\displaystyle{ \scriptstyle\mathcal{A} }[/math] ∩ τ,

[math]\displaystyle{ \mu\left(\bigcup \mathcal{G} \right) = \sup_{G\in\mathcal{G}} \mu(G) }[/math],

we say that μ is τ-additive. In particular, if μ is inner regular (with respect to compact sets) then it is τ-additive.

Properties

Basic properties

Useful properties of an additive function μ include the following:

1. Either μ(∅) = 0, or μ assigns ∞ to all sets in its domain, or μ assigns −∞ to all sets in its domain.
2. If μ is non-negative and A ⊆ B, then μ(A) ≤ μ(B).
3. If A ⊆ B and μ(B) − μ(A) is defined, then μ(B \ A) = μ(B) − μ(A).
4. Given A and B, μ(A ∪ B) + μ(A ∩ B) = μ(A) + μ(B).

Examples

An example of a σ-additive function is the function μ defined over the power set of the real numbers, such that

[math]\displaystyle{ \mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\ 0 & \mbox{ if } 0 \notin A. \end{cases} }[/math]

If [math]\displaystyle{ A_1,A_2,\dots,A_n,\dots }[/math] 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

[math]\displaystyle{ \mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n) }[/math]

holds.

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

An additive function which is not σ-additive

An example of an additive function which is not σ-additive is obtained by considering μ, defined over the Lebesgue sets of the real numbers by the formula

[math]\displaystyle{ \mu(A)=\lim_{k\to\infty} \frac{1}{k} \cdot \lambda\left(A \cap \left(0,k\right)\right), }[/math]

where λ denotes the Lebesgue measure and lim the Banach limit.

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

[math]\displaystyle{ A_n=\left[n,n+1\right) }[/math]

for n=0, 1, 2, ... The union of these sets is the positive reals, and μ applied to the union is then one, while μ applied to any of the individual sets is zero, so the sum of μ(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

• signed measure
• measure (mathematics)
• additive map
• subadditive function
• σ-finite measure
• Hahn–Kolmogorov theorem
• τ-additivity