# Content (measure theory)

In mathematics, a content is a set function that is like a measure, but a content must only be finitely additive, whereas a measure must be countably additive. A content is a real function ${\displaystyle \mu }$ 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. ${\displaystyle \mu (A_{1}\cup A_{2})=\mu (A_{1})+\mu (A_{2}){\text{ whenever }}A_{1},A_{2},A_{1}\cup A_{2}\ \in {\mathcal {A}}{\text{ and }}A_{1}\cap A_{2}=\varnothing .}$

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.