Sigma additivity


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.

Contents

Additive (or finitely additive) set functions

Let μ be a function defined on an algebra of sets \scriptstyle\mathcal{A} with values in [−∞, +∞] (see the extended real number line). The function μ is called 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 \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:

  1. μ(∅) = 0.
  2. If μ is non-negative and AB, then μ(A) ≤ μ(B).
  3. If AB, then μ(B - A) = μ(B) - μ(A).
  4. Given A and B, μ(AB) + μ(AB) = 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+1},\,  \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(An) 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

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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Sigma-algebra — In mathematics, a σ algebra (also sigma algebra, σ field, sigma field) is a technical concept for a collection of sets satisfying certain properties. The main use of σ algebras is in the definition of measures; specifically, the collection of… …   Wikipedia

  • Hahn–Kolmogorov theorem — In mathematics, the Hahn–Kolmogorov theorem characterizes when a finitely additive function with non negative (possibly infinite) values can be extended to a bona fide measure. It is named after the Austrian mathematician Hans Hahn and the… …   Wikipedia

  • Hahn decomposition theorem — In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space ( X , Sigma;) and a signed measure mu; defined on the sigma; algebra Sigma;, there exist two sets P and N in… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Probability axioms — In probability theory, the probability P of some event E , denoted P(E), is defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov.These assumptions can be summarised as: Let (Ω, F , P ) be a measure space… …   Wikipedia

  • Egorov's theorem — In measure theory, an area of mathematics, Egorov s theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. The theorem is named after Dmitri Egorov, a Russian physicist and geometer …   Wikipedia

  • Additive function — Different definitions exist depending on the specific field of application. Traditionally, an additive function is a function that preserves the addition operation:: f ( x + y ) = f ( x ) + f ( y )for any two elements x and y in the domain. An… …   Wikipedia

  • Measure (mathematics) — Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. In mathematical analysis …   Wikipedia

  • Calculation of glass properties — [ refractive index. [ [http://glassproperties.com/refractive index/ Calculation of the Refractive Index of Glasses] ] ] The calculation of glass properties (glass modeling) is used to predict glass properties of interest or glass behavior under… …   Wikipedia

  • Analysis of variance — In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of… …   Wikipedia