Sigma additivity

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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[edit]

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[edit]

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.

Properties[edit]

Basic properties[edit]

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 AB, then μ(A) ≤ μ(B).
  3. If AB and μ(B) − μ(A) is defined, then μ(B \ A) = μ(B) − μ(A).
  4. Given A and B, μ(AB) + μ(AB) = μ(A) + μ(B).

Examples[edit]

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[edit]

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} k \cdot \lambda\left(A \cap \left(0,\frac{1}{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[\frac {1}{n+1},\,  \frac{1}{n}\right)

for n=1, 2, 3, ... The union of these sets is the interval (0, 1), 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(An) is also zero, which proves the counterexample.

Generalizations[edit]

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[edit]

This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.