In
mathematics, an additive set function is a
function mapping sets to numbers, with the property that its value on a
union of two
disjoint sets equals the sum of its values on these sets, namely, If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive
set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for
countably infinite many sets, that is,
Additivity and sigma-additivity are particularly important properties of
measures. They 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; that is, σ-additivity implies additivity.
Suppose that is a
σ-algebra. If for every
sequence of pairwise disjoint sets in
holds then is said to be countably additive or 𝜎-additive.
Every 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 If for every
directed family of measurable
open sets
we say that is -additive. In particular, if is
inner regular (with respect to compact sets) then it is τ-additive.[1]
Properties
Useful properties of an additive set function include the following.
Value of empty set
Either or assigns to all sets in its domain, or assigns to all sets in its domain. Proof: additivity implies that for every set If then this equality can be satisfied only by plus or minus infinity.
Monotonicity
If is non-negative and then That is, is a monotone set function. Similarly, If is non-positive and then
Note that modularity has a different and unrelated meaning in the context of complex functions; see
modular form.
Set difference
If and is defined, then
Examples
An example of a 𝜎-additive function is the function 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
A charge is defined to be a finitely additive set function that maps to [2] (Cf.
ba space for information about bounded charges, where we say a charge is bounded to mean its range is a bounded subset of R.)
An additive function which is not σ-additive
An example of an additive function which is not σ-additive is obtained by considering , defined over the Lebesgue sets of the
real numbers by the formula
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 The union of these sets is the
positive reals, and applied to the union is then one, while applied to any of the individual sets is zero, so the sum of is also zero, which proves the counterexample.