Content (measure theory)

In mathematics, in particular in measure theory, a content ${\displaystyle \mu }$ is a real-valued function defined on a collection of subsets ${\displaystyle {\mathcal {A}}}$ such that

1. ${\displaystyle \mu (A)\in \ [0,\infty ]{\text{ whenever }}A\in {\mathcal {A}}.}$
2. ${\displaystyle \mu (\varnothing )=0.}$
3. ${\displaystyle \mu {\Bigl (}\bigcup _{i=1}^{n}A_{i}{\Bigr )}=\sum _{i=1}^{n}\mu (A_{i}){\text{ whenever }}A_{1},\dots ,A_{n},\bigcup _{i=1}^{n}A_{i}\in {\mathcal {A}}{\text{ and }}A_{i}\cap A_{j}=\varnothing {\text{ for }}i\neq j.}$

That is, a content is a generalization of a measure: while the latter must be countably additive, the former must only be finitely additive.

In many important applications the ${\displaystyle {\mathcal {A}}}$ is chosen to be a ring of sets or to be at least a semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings.

If a content is additionally σ-additive it is called a pre-measure and if furthermore ${\displaystyle {\mathcal {A}}}$ is a σ-algebra, the content is called a measure. Therefore, every (real-valued) measure is a content, but not vice versa. Contents give a good notion of integrating bounded functions on a space but can behave badly when integrating unbounded functions, while measures give a good notion of integrating unbounded functions.