Pre-measure
# Pre-measure

In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.

## Definition

Families ${\mathcal {F}}$ of sets over $\Omega$ vte :

| Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under: | Directed by $\,\supseteq$ | $A\cap B$ | $A\cup B$ | $B\setminus A$ | $\Omega \setminus A$ | $A_{1}\cap A_{2}\cap \cdots$ | $A_{1}\cup A_{2}\cup \cdots$ | $\Omega \in {\mathcal {F}}$ | $\varnothing \in {\mathcal {F}}$ | F.I.P. |
| - | - | - | - | - | - | - | - | - | - | - |
| π-system | | | | | | | | | | |
| Semiring | | | | | | | | | | Never |
| Semialgebra (Semifield) | | | | | | | | | | Never |
| Monotone class | | | | | | only if $A_{i}\searrow$ | only if $A_{i}\nearrow$ | | | |
| 𝜆-system (Dynkin System) | | | | only if $A\subseteq B$ | | | only if $A_{i}\nearrow$ or they are disjoint | | | Never |
| Ring (Order theory) | | | | | | | | | | |
| Ring (Measure theory) | | | | | | | | | | Never |
| δ-Ring | | | | | | | | | | Never |
| 𝜎-Ring | | | | | | | | | | Never |
| Algebra (Field) | | | | | | | | | | Never |
| 𝜎-Algebra (𝜎-Field) | | | | | | | | | | Never |
| Dual ideal | | | | | | | | | | |
| Filter | | | | Never | Never | | | | $\varnothing \not \in {\mathcal {F}}$ | |
| Prefilter (Filter base) | | | | Never | Never | | | | $\varnothing \not \in {\mathcal {F}}$ | |
| Filter subbase | | | | Never | Never | | | | $\varnothing \not \in {\mathcal {F}}$ | |
| Open Topology | | | | | | | (even arbitrary $\cup$) | | | Never |
| Closed Topology | | | | | | (even arbitrary $\cap$) | | | | Never |
| Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under: | directed downward | finite intersections | finite unions | relative complements | complements in $\Omega$ | countable intersections | countable unions | contains $\Omega$ | contains $\varnothing$ | Finite Intersection Property |
| Additionally, a semiring is a π-system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$ A semialgebra is a semiring that contains $\Omega .$ $A,B,A_{1},A_{2},\ldots$ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$ |||||||||||

Let $R$ be a ring of subsets (closed under union and relative complement) of a fixed set $X$ and let $\mu _{0}:R\to [0,\infty ]$ be a set function. $\mu _{0}$ is called a pre-measure if

$\mu _{0}(\varnothing )=0$

and, for every countable (or finite) sequence $A_{1},A_{2},\ldots \in R$ of pairwise disjoint sets whose union lies in $R,$

$\mu _{0}\left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu _{0}(A_{n}).$

The second property is called $\sigma$-additivity.

Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra (or a sigma-ring).

## Carathéodory's extension theorem

It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space $X.$ More precisely, if $\mu _{0}$ is a pre-measure defined on a ring of subsets $R$ of the space $X,$ then the set function $\mu ^{*}$ defined by

$\mu ^{*}(S)=\inf \left\{\left.\sum _{i=1}^{\infty }\mu _{0}(A_{i})\right|A_{i}\in R,S\subseteq \bigcup _{i=1}^{\infty }A_{i}\right\}$

is an outer measure on $X$ and the measure $\mu$ induced by $\mu ^{*}$ on the $\sigma$-algebra $\Sigma$ of Carathéodory-measurable sets satisfies $\mu (A)=\mu _{0}(A)$ for $A\in R$ (in particular, $\Sigma$ includes $R$). The infimum of the empty set is taken to be $+\infty .$

(Note that there is some variation in the terminology used in the literature. For example, Rogers (1998) uses "measure" where this article uses the term "outer measure". Outer measures are not, in general, measures, since they may fail to be $\sigma$-additive.)

