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Concept
Set function
Summary
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers 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.
If is a family of sets over (meaning that where denotes the powerset) then a is a function with domain and codomain or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures.
The domain of a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below.
In general, it is typically assumed that is always well-defined for all or equivalently, that does not take on both and as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever is finitely additive:
is defined with satisfying and
Null sets
A set is called a (with respect to ) or simply if
Whenever is not identically equal to either or then it is typically also assumed that:
if
Variation and mass
The is
where denotes the absolute value (or more generally, it denotes the norm or seminorm if is vector-valued in a (semi)normed space).
Assuming that then is called the of and is called the of
A set function is called if for every the value is (which by definition means that and ; an is one that is equal to or ).
Every finite set function must have a finite mass.
Official source
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