Set function

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 ${\displaystyle \mathbb {R} \cup \{\pm \infty \},}$ which consists of the real numbers ${\displaystyle \mathbb {R} }$ and ${\displaystyle \pm \infty .}$

A set function generally aims to measure 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 ${\displaystyle {\mathcal {F}}}$ is a family of sets over ${\displaystyle \Omega }$ (meaning that ${\displaystyle {\mathcal {F}}\subseteq \wp (\Omega )}$ where ${\displaystyle \wp (\Omega )}$ denotes the powerset) then a set function on ${\displaystyle {\mathcal {F}}}$ is a function ${\displaystyle \mu }$ with domain ${\displaystyle {\mathcal {F}}}$ and codomain ${\displaystyle [-\infty ,\infty ]}$ 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 ${\displaystyle \mu (E)+\mu (F)}$ is always well-defined for all ${\displaystyle E,F\in {\mathcal {F}},}$ or equivalently, that ${\displaystyle \mu }$ does not take on both ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$ 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 ${\displaystyle \mu }$ is finitely additive:

Null sets

A set ${\displaystyle F\in {\mathcal {F}}}$ is called a null set (with respect to ${\displaystyle \mu }$) or simply null if ${\displaystyle \mu (F)=0.}$ Whenever ${\displaystyle \mu }$ is not identically equal to either ${\displaystyle -\infty }$ or ${\displaystyle +\infty }$ then it is typically also assumed that:

Variation and mass

The total variation of a set ${\displaystyle S}$ is $${\displaystyle |\mu |(S)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\sup\{|\mu (F)|:F\in {\mathcal {F}}{\text{ and }}F\subseteq S\}}$$ where ${\displaystyle |\,\cdot \,|}$ denotes the absolute value (or more generally, it denotes the norm or seminorm if ${\displaystyle \mu }$ is vector-valued in a (semi)normed space). Assuming that ${\displaystyle \cup {\mathcal {F}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\textstyle \bigcup \limits _{F\in {\mathcal {F}}}F\in {\mathcal {F}},}$ then ${\displaystyle |\mu |\left(\cup {\mathcal {F}}\right)}$ is called the total variation of ${\displaystyle \mu }$ and ${\displaystyle \mu \left(\cup {\mathcal {F}}\right)}$ is called the mass of ${\displaystyle \mu .}$