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.
The term modular set function is equivalent to additive set function; see modularity below.
(see the extended real number line).
One can prove by mathematical induction that an additive function satisfies
Every 𝜎-additive function is additive but not vice versa, as shown below.
Suppose that in addition to a sigma algebra
If for every directed family of measurable open sets
is inner regular (with respect to compact sets) then it is τ-additive.
[1] Useful properties of an additive set function
Proof: additivity implies that for every set
(it's possible in the edge case of an empty domain that the only choice for
then this equality can be satisfied only by plus or minus infinity.
is called a modular set function and a valuation if whenever
However, the related properties of submodularity and subadditivity are not equivalent to each other.
defined over the power set of the real numbers, such that
A charge is defined to be a finitely additive set function that maps
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 example of an additive function which is not σ-additive is obtained by considering
, defined over the Lebesgue sets of the real numbers
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
The union of these sets is the positive reals, and
applied to any of the individual sets is zero, so the sum of
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.