Sigma additivity

In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size (length, area, volume) of a set.

1 Additive (or finitely additive) set functions
2 σ-additive set functions
3 Properties
- 3.1 Basic properties
4 Examples
5 Generalizations
6 See also

Additive (or finitely additive) set functions

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

$\mu(A \cup B) = \mu(A) %2B \mu(B) .$

(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

$\mu\left(\bigcup_{n=1}^N A_n\right)=\sum_{n=1}^N \mu(A_n)$

for any $A_1,A_2,\dots,A_N$ disjoint sets in $\scriptstyle\mathcal{A}$ .

σ-additive set functions

Suppose $\scriptstyle\mathcal{A}$ is a σ-algebra. If for any sequence $A_1,A_2,\dots,A_k,\dots$ of disjoint sets in $\scriptstyle\mathcal{A}$ one has

$\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n),$

we say that μ is countably additive or σ-additive.

Any σ-additive function is additive but not vice-versa, as shown below.

Properties

Basic properties

Useful properties of an additive function μ include the following:

μ(∅) = 0.
If μ is non-negative and A ⊆ B, then μ(A) ≤ μ(B).
If A ⊆ B, then μ(B - A) = μ(B) - μ(A).
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

$\mu (A)= \begin{cases} 1 & \mbox{ if } 0 \in A \\ 0 & \mbox{ if } 0 \notin A. \end{cases}$

If $A_1,A_2,\dots,A_k,\dots$ 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

$\mu\left(\bigcup_{n=1}^\infty A_n\right) = \sum_{n=1}^\infty \mu(A_n)$

holds.

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

An example of an additive function which is not σ-additive is obtained by considering μ, defined over the power set of the real numbers by the slightly modified formula

$\mu (A)= \begin{cases} \infty & \mbox { if } 0 \in \bar A \\ 0 & \mbox { if } 0 \notin \bar A \end{cases}$

where the bar denotes the closure of a set.

One can check that this function is additive by using the property that the closure of a finite union of sets is the union of the closures of the sets, and looking at the cases when 0 is in the closure of any of those sets or not. That this function is not σ-additive follows by considering the sequence of disjoint sets

$A_n=\left[\frac {1}{n%2B1},\, \frac{1}{n}\right)$

for n=1, 2, 3, ... The union of these sets is the interval (0, 1) whose closure is [0, 1] and μ applied to the union is then infinity, 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.