Set function

In mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input is a set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.

Examples

Examples of set functions include:

• The function that assigns to each set its cardinality, i.e. the number of members of the set, is a set function.
• The function

$d(A)=\lim _{n\to \infty }{\frac {|A\cap \{1,\dots ,n\}|}{n}},$

assigning densities to sufficiently well-behaved subsets A ⊆ {1, 2, 3, ...}, is a set function.

• The Lebesgue measure is a set function that assigns a non-negative real number to any set of real numbers, that is in Lebesgue $\sigma$-algebra. (Kolmogorov and Fomin 1975)
• A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the empty set is zero and the probability of the sample space is 1, with other sets given probabilities between 0 and 1.
• A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory.
• A random set is a set-valued random variable. See Random compact set.

References

• A.N. Kolmogorov and S.V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0

Further reading

• Sobolev, V.I. (2001) [1994], "Set function", Encyclopedia of Mathematics, EMS Press
• *Regular set function at Encyclopedia of Mathematics