Sigma additivity

In mathematics, additivity and sigma additivity (also called countable 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.

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, or finitely additive, if, whenever A and B are disjoint sets in $\scriptstyle\mathcal{A}$ , one has

\mu(A \cup B) = \mu(A) + \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 that $\scriptstyle\mathcal{A}$ is a σ-algebra. If for any sequence $A_1,A_2,\dots,A_n,\dots$ 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.

τ-additive set functions

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

\mu\left(\bigcup \mathcal{G} \right) = \sup_{G\in\mathcal{G}} \mu(G)

,

we say that μ is τ-additive. In particular, if μ is inner regular then it is τ-additive.^[1]

Properties

Basic properties

Useful properties of an additive function μ include the following:

Either μ(∅) = 0, or μ assigns ∞ to all sets in its domain, or μ assigns −∞ to all sets in its domain.
If μ is non-negative and A ⊆ B, then μ(A) ≤ μ(B).
If A ⊆ B and μ(B) − μ(A) is defined, 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 $A_1,A_2,\dots,A_n,\dots$ 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 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

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

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

A_n=\left[n,n+1\right)

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.

References

↑ D.H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.

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