Content (measure theory)

If ${\displaystyle {\mathcal {A}}}$ forms a Semiring of sets then the following statements can be deduced:

• Every content ${\displaystyle \mu }$ is monotone that is,
  $${\displaystyle A\subseteq B\Rightarrow \mu (A)\leq \mu (B){\text{ for }}A,B\in {\mathcal {A}}.}$$

  
• Every content ${\displaystyle \mu }$ is subadditive that is,

${\displaystyle \mu (A\cup B)\leq \mu (A)+\mu (B)}$ for ${\displaystyle A,B\in {\mathcal {A}}}$ such that ${\displaystyle A\cup B\in {\mathcal {A}}.}$

If furthermore ${\displaystyle {\mathcal {A}}}$ is a Ring of sets one gets additionally:

• Subtractivity: for ${\displaystyle B\subseteq A}$ satisfying ${\displaystyle \mu (B)<\infty }$ it follows ${\displaystyle \mu (A\setminus B)=\mu (A)-\mu (B).}$
• ${\displaystyle A,B\in {\mathcal {A}}\Rightarrow \mu (A\cup B)+\mu (A\cap B)=\mu (A)+\mu (B).}$
• Subadditivity: ${\displaystyle A_{i}\in {\mathcal {A}}\;(i=1,2,\dotsc ,n)\Rightarrow \mu \left(\bigcup _{i=1}^{n}A_{i}\right)\leq \sum _{i=1}^{n}\mu (A_{i}).}$
• ${\displaystyle \sigma }$-Superadditivity: For any ${\displaystyle A_{i}\in {\mathcal {A}}\;(i=1,2,\dotsc )\ }$ pairwise disjoint satisfying ${\displaystyle \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {A}}}$ we have ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)\geq \sum _{i=1}^{\infty }\mu (A_{i}).}$
• If ${\displaystyle \mu }$ is a finite content, that is, ${\displaystyle A\in {\mathcal {A}}\Rightarrow \mu (A)<\infty ,}$ then the inclusion–exclusion principle applies:
  $${\displaystyle \mu \left(\bigcup _{i=1}^{n}A_{i}\right)=\sum _{k=1}^{n}(-1)^{k+1}\!\!\sum _{I\subseteq \{1,\dotsc ,n\}, \atop |I|=k}\!\!\!\!\mu \left(\bigcap _{i\in I}A_{i}\right)}$$

  
  where ${\displaystyle A_{i}\in {\mathcal {A}}}$ for all ${\displaystyle i\in \{1,\dotsc ,n\}.}$

Given λ as above, we define a function μ on all open sets by

$${\displaystyle \mu (U)=\sup _{C\subseteq U}\lambda (C).}$$

This has the following properties:

1. ${\displaystyle \mu (U)\in \ [0,\infty ]}$
2. ${\displaystyle \mu (\varnothing )=0}$
3. ${\displaystyle \mu (U_{1})\leq \mu (U_{2}){\text{ whenever }}U_{1}\subseteq U_{2}}$
4. ${\displaystyle \mu \left(\bigcup _{n}U_{n}\right)\leq \sum _{n}\lambda (U_{n})}$ for any collection of open sets
5. ${\displaystyle \mu \left(\bigcup _{n}U_{n}\right)=\sum _{n}\lambda (U_{n})}$ for any collection of disjoint open sets.

Given μ as above, we extend the function μ to all subsets of the topological space by

$${\displaystyle \mu (A)=\inf _{A\subseteq U}\mu (U).}$$

This is an outer measure, in other words it has the following properties:

1. ${\displaystyle \mu (A)\in \ [0,\infty ]}$
2. ${\displaystyle \mu (\varnothing )=0.}$
3. ${\displaystyle \mu (A_{1})\leq \mu (A_{2}){\text{ whenever }}A_{1}\subseteq A_{2}}$
4. ${\displaystyle \mu \left(\bigcup _{n}A_{n}\right)\leq \sum _{n}\lambda (A_{n})}$ for any countable collection of sets.

The function μ above is an outer measure on the family of all subsets. Therefore it becomes a measure when restricted to the measurable subsets for the outer measure, which are the subsets ${\displaystyle E}$ such that ${\displaystyle \mu (X)=\mu (X\cap E)+\mu (X\setminus E)}$ for all subsets ${\displaystyle X.}$ If the space is locally compact then every open set is measurable for this measure.

The measure ${\displaystyle \mu }$ does not necessarily coincide with the content ${\displaystyle \lambda }$ on compact sets, However it does if ${\displaystyle \lambda }$ is regular in the sense that for any compact ${\displaystyle C,}$ ${\displaystyle \lambda (C)}$ is the inf of ${\displaystyle \lambda (D)}$ for compact sets ${\displaystyle D}$ containing ${\displaystyle C}$ in their interiors.