 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 with values in [−∞, +∞] (see the extended real number line). The function μ is called 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 .
σadditive set functions
Suppose is a σalgebra. If for any sequence of disjoint sets in one has
we say that μ is countably additive or σadditive.
Any σadditive function is additive but not viceversa, as shown below.
Properties
Basic properties
Useful properties of an additive function μ include the following:
 μ(∅) = 0.
 If μ is nonnegative 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 σadditive 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 σadditive functions.
An example of an additive function which is not σadditive is obtained by considering μ, defined over the power set of the real numbers by the slightly modified formula
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 σadditive follows by considering the sequence of disjoint sets
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 sigmaadditivity, one needs in addition that the concept of limit of a sequence be defined on that set. For example, spectral measures are sigmaadditive functions with values in a Banach algebra. Another example, also from quantum mechanics, is the positive operatorvalued measure.
See also
 signed measure
 measure (mathematics)
 additive function
 subadditive function
 HahnKolmogorov theorem
This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
Categories: Measure theory
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