Sigma additivity


In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of how intuitive properties of size 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 functions

Let be a function defined on an algebra of sets with values in . The function is called additive, or finitely additive, if, whenever A and B are disjoint sets in , one has
°
One can prove by mathematical induction that an additive function satisfies
for any disjoint sets in.

σ-additive set functions

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 functions

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 then it is τ-additive.

Properties

Basic properties

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 A ⊆ B, then μ ≤ μ.
  3. If A ⊆ B and μ − μ is defined, then μ = μ − μ.
  4. Given A and B, μ + μ = μ + μ.

    Examples

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-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
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 is also zero, which proves the counterexample.

Generalizations

One may define additive functions with values in any additive monoid. 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.