Content (measure theory)

In measure theory, content is a special set function that is defined for certain set systems and serves to abstract and generalize the intuitive concept of volume.

definition

Finite additivity for a content : The content of a finitely disjoint union is equal to the sum of the contents of the individual subsets. ${\displaystyle \mu }$

On any quantity system

A set system is given that contains the empty set. Then it is called a set function ${\displaystyle {\mathcal {C}}}$

${\displaystyle \mu :{\mathcal {C}}\to [0,\infty ]}$

a content if:

1. The empty set has zero value: . ${\displaystyle \mu (\emptyset )=0}$
2. The function is finite additive . So if there are finitely many pairwise disjoint sets and then we have ${\displaystyle A_{1},A_{2},\dotsc ,A_{n}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \textstyle \bigcup _{i=1}^{n}A_{i}\in {\mathcal {C}}}$

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

The system of quantities is usually a half- ring of quantities .

comment

It should be noted that the definition does not require that finite unions of disjoint sets are again in the set system. It is only required that if the disjoint union is again in the set system, the finite additivity holds. For example, finite unions of disjoint sets in half-rings are generally not in the half-ring again. An example of this is the half-ring on , which consists of the half-open intervals of the shape . ${\displaystyle \mathbb {R} }$ ${\displaystyle [a,b)}$

Likewise, it generally follows from additivity, i.e. from the property

${\displaystyle \mu (A\cup B)=\mu (A)+\mu (B)}$

for disjoint sets with not finite additivity. This is based on the fact that in general set systems it does not follow for disjoints . The (backwards) inductive conclusion from additivity to finite additivity is only valid in unification- stable set systems. ${\displaystyle A,B}$ ${\displaystyle A\cup B\in {\mathcal {C}}}$ ${\displaystyle A\cup B\in {\mathcal {C}}}$ ${\displaystyle A\cup B\cup C\in {\mathcal {C}}}$ ${\displaystyle C\in {\mathcal {C}}}$

On unification-stable quantity systems

Based on the above considerations, the following simplified definition is obtained in unification-stable set systems: If a unification- stable set system that contains empty sets, then it is called a set function ${\displaystyle {\mathcal {V}}}$

${\displaystyle \mu :{\mathcal {V}}\to [0,\infty ]}$

a content if:

1. The empty set has zero value: . ${\displaystyle \mu (\emptyset )=0}$
2. The function is additive , i.e. it holds for every two disjoint sets ${\displaystyle A,B\in {\mathcal {V}}}$

   ${\displaystyle \mu (A\cup B)=\mu (A)+\mu (B)}$.

The unification-stable set systems are mostly a set ring .

Examples

The most important content is the so-called Lebesgue content

${\displaystyle \mu ([a,b))=b-a}$.

on the half-ring of the half-open intervals on the real numbers. The Lebesgue integral is finally constructed from it through expansion and various continuation sentences. In fact, this content is already a premeasure . ${\displaystyle [a,b)}$

Another important content is the Stieltjes content , from which the Lebesgue-Stieltjes measure and the Lebesgue-Stieltjes integral are derived:

${\displaystyle \mu _{F}([a,b))=F(b)-F(a)}$,

where is a monotonically increasing real-valued function . It can be used to describe all finite contents on the real numbers. ${\displaystyle F}$

Another content is the Jordan measure . Contrary to the name, it is not a measure in the sense of measure theory.

properties

Depending on the set system on which content is defined, certain properties apply.

In a half ring

If is a half ring , then: ${\displaystyle {\mathcal {C}}={\mathcal {H}}}$

• Every content is monotonous , so the following applies: ${\displaystyle \mu }$

${\displaystyle A\subseteq B\Rightarrow \mu (A)\leq \mu (B)}$for . ${\displaystyle A,B\in {\mathcal {H}}}$

• All content is subadditive , so the following applies: ${\displaystyle \mu }$

${\displaystyle \mu (A\cup B)\leq \mu (A)+\mu (B)}$for off with . ${\displaystyle A,B}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle A\cup B\in {\mathcal {H}}}$

In the ring

If you choose a ring as the system of quantities, the following statements apply in addition to the properties in the half ring (since each ring is a half ring):

• Subtraktivität : for with true . ${\displaystyle B\subseteq A}$ ${\displaystyle \mu (B)<\infty }$ ${\displaystyle \mu (A\setminus B)=\mu (A)-\mu (B)}$
• ${\displaystyle A,B\in {\mathcal {R}}\Rightarrow \mu (A\cup B)+\mu (A\cap B)=\mu (A)+\mu (B)}$.
• Subadditivity : . ${\displaystyle A_{i}\in {\mathcal {R}}\;(i=1,2,\dotsc ,n)\Rightarrow \mu \left(\bigcup _{i=1}^{n}A_{i}\right)\leq \sum _{i=1}^{n}\mu (A_{i})}$
• ${\displaystyle \sigma }$-Superadditivity : be pairwise disjoint with . Then it follows from the additivity and monotony . ${\displaystyle A_{i}\in {\mathcal {R}}\;(i=1,2,\dotsc )\ }$ ${\displaystyle \bigcup _{i=1}^{\infty }A_{i}\in {\mathcal {R}}}$ ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)\geq \sum _{i=1}^{\infty }\mu (A_{i})}$
• If is finite, i.e. applies to all , then the sieve formula of Poincaré and Sylvester applies : ${\displaystyle \mu }$ ${\displaystyle A\in {\mathcal {R}}\Rightarrow \mu (A)<\infty }$

${\displaystyle \mu \left(\bigcup _{i=1}^{n}A_{i}\right)=\sum _{k=1}^{n}(-1)^{k+1}\!\!\sum _{I\subseteq \{1,\dotsc ,n\}, \atop |I|=k}\!\!\!\!\mu \left(\bigcap _{i\in I}A_{i}\right)}$

with for . ${\displaystyle A_{i}\in {\mathcal {C}}}$ ${\displaystyle i\in \{1,\dotsc ,n\}}$

Derived terms

A content is finite if it applies to everyone . A content is called σ-finite if there is a decomposition of in such that holds for all . ${\displaystyle \mu (A)<\infty }$ ${\displaystyle A\in {\mathcal {C}}}$ ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mu (A_{i})<\infty }$ ${\displaystyle i\in \mathbb {N} }$

Continuation of content

For every content on the half - ring, you can construct a content on the ring generated by . Due to the properties of a half ring, there are pairwise disjoint sets with for all . By going through ${\displaystyle \mu }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle \mu '}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {R}}}$ ${\displaystyle A\in {\mathcal {R}}}$ ${\displaystyle A_{1},A_{2},\dotsc ,A_{m}\in {\mathcal {H}}}$ ${\displaystyle \textstyle A=\bigcup _{j=1}^{m}A_{j}}$ ${\displaystyle \mu '}$

${\displaystyle \mu '(A):=\sum _{j=1}^{m}\mu (A_{j})}$

defined, you get a clearly defined continuation . The continuation is -finite exactly if -finite. ${\displaystyle \mu '}$ ${\displaystyle \mu '}$ ${\displaystyle \sigma }$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$

Related concepts

Probability content

A content is called a probability content if the basic set is contained in the set system and is valid. ${\displaystyle \mu }$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mu (\Omega )=1}$

Signed content

A signed content is a set function on a set system that is closed with respect to finite unions and contains the empty set for which applies ${\displaystyle \nu }$ ${\displaystyle {\mathcal {M}}}$

1. ${\displaystyle \nu (\emptyset )=0}$
2. The image set of the set function is or . ${\displaystyle [-\infty ,+\infty )}$ ${\displaystyle (-\infty ,+\infty ]}$
3. Finite additivity applies, i.e. for disjoint . ${\displaystyle \nu (A\cup B)=\nu (A)+\nu (B)}$ ${\displaystyle A,B\in {\mathcal {M}}}$

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ^ Schmidt: Measure and probability. 2011, p. 44.
2. ↑ Klenke: Probability Theory. 2013, p. 12.
3. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 27.
4. ^ Schmidt: Measure and Probability. 2011, p. 194.
5. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 277.