In mathematics, especially
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, a set function is a
function whose
domain is a
family
Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
of
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of some given set and that (usually) takes its values in the
extended real number line which consists of the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
and
A set function generally aims to subsets in some way.
Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
Definitions
If
is a
family of sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of su ...
over
(meaning that
where
denotes the
powerset) then a is a function
with
domain and
codomain