Sigma additivity

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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, and σ-additivity implies additivity.

Additive (or finitely additive) set functionsEdit

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

 

(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

 

for any   disjoint sets in  .

σ-additive set functionsEdit

Suppose that   is a σ-algebra. If for any sequence   of pairwise disjoint sets in  , one has

 ,

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 functionsEdit

Suppose that in addition to a sigma algebra  , we have a topology τ. If for any directed family of measurable open sets   ∩τ,

 ,

we say that μ is τ-additive. In particular, if μ is inner regular (with respect to compact sets) then it is τ-additive.[1]

PropertiesEdit

Basic propertiesEdit

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).

ExamplesEdit

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

 

If   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

 

holds.

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

An additive function which is not σ-additiveEdit

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

 

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

 

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(An) is also zero, which proves the counterexample.

GeneralizationsEdit

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 alsoEdit

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

ReferencesEdit

  1. ^ D.H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003.