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

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

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

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

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

The second property is called ${\displaystyle \sigma }$-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

Main article: 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 ${\displaystyle X.}$ More precisely, if ${\displaystyle \mu _{0}}$ is a pre-measure defined on a ring of subsets ${\displaystyle R}$ of the space ${\displaystyle X,}$ then the set function ${\displaystyle \mu ^{*}}$ defined by

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

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

(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 ${\displaystyle \sigma }$-additive.)

See also

• Hahn-Kolmogorov theorem

References

• Munroe, M. E. (1953). Introduction to measure and integration. Cambridge, Mass.: Addison-Wesley Publishing Company Inc. p. 310. MR 0053186
• Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. p. 195. ISBN   0-521-62491-6. MR 1692618 (See section 1.2.)
• Folland, G. B. (1999). Real Analysis . Pure and Applied Mathematics (Second ed.). New York: John Wiley & Sons, Inc. pp.  30–31. ISBN   0-471-31716-0.