σ-finiteness

The concept of -Endlichkeit (also -Finitheit ) is in mathematical measure theory used and provides a gradation of (measurable) amounts of infinite extent in -endliche and not -endliche amounts. It is introduced for similar reasons as the concept of countability in relation to the number of elements in a set. In general, the finiteness is a property of set functions in connection with a set system . Often, however, the system of quantities is not given if it is clear what it is. ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Definition of dimensions

A measuring room is given . Then a measure is called a -finite measure if it fulfills one of the following three equivalent conditions: ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$

1. There are more than a countable number of sets of which also for all meet and overlap. So it applies ${\displaystyle A_{1},A_{2},A_{3},\dots }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mu (A_{n})<\infty }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle X}$

   ${\displaystyle \bigcup _{n\in \mathbb {N} }A_{n}=X}$.

2. There are more than a countable number of disjoint sets of which also for all meet and overlap. So it applies ${\displaystyle A_{1},A_{2},A_{3},\dots }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mu (A_{n})<\infty }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle X}$

   ${\displaystyle \bigcup _{n\in \mathbb {N} }A_{n}=X}$.

3. There is a strictly positive (i.e. for all ) measurable function such that ${\displaystyle f(x)>0}$ ${\displaystyle x\in X}$ ${\displaystyle f}$

   ${\displaystyle \int f(x)\mu (\mathrm {d} x)<\infty }$.

The dimension space is then also referred to as the -finite dimension space . More generally, a signed measure is called -finite if its variation is -finite. ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Examples

The Lebesgue measure on the real numbers, provided with Borel's σ-algebra , is not finite, but -finite. Because if you look at the quantities ${\displaystyle \lambda }$ ${\displaystyle \sigma }$

${\displaystyle I_{n}:=(-n,n)\in {\mathcal {B}}(\mathbb {R} )}$,

so is and ${\displaystyle \lambda (I_{n})=2n<\infty }$

${\displaystyle \bigcup _{n=1}^{\infty }I_{n}=\mathbb {R} }$.

Thus the Lebesgue measure fulfills the first criterion in the above construction. A disjoint coverage with sets of finite measure as in the second point of the definition is provided by the sets, for example

${\displaystyle B_{n}=I_{n}\setminus I_{n-1}}$,

where is. Then it is and it applies again ${\displaystyle B_{1}=I_{1}}$ ${\displaystyle \lambda (B_{n})=2}$

${\displaystyle \bigcup _{n=1}^{\infty }B_{n}=\mathbb {R} }$.

A strictly positive function with a finite integral as required in the third point of the definition is obtained, for example, from

${\displaystyle f(x)=\sum _{n=1}^{\infty }{\frac {\mathbf {1} _{B_{n}}(x)}{2^{n}}}}$.

Here the indicator function is on the crowd . ${\displaystyle \mathbf {1} _{A}}$ ${\displaystyle A}$

It should be noted that -finitude is always a property of a measure in combination with a measuring space. So the counting measure on a set , provided with the power set as -algebra, is finite if is finite and -final if and only if it is at most countable. ${\displaystyle \sigma }$ ${\displaystyle M}$ ${\displaystyle \sigma }$ ${\displaystyle M}$ ${\displaystyle \sigma }$ ${\displaystyle M}$

application

• Infinite measures can have pathological properties, but many of the frequently considered measures are not finite. The class of -finite measures shares some pleasant properties with finite measures , -finiteness can be compared in this respect with the separability of topological spaces . For example, some theorems of analysis, such as Radon-Nikodým's theorem and Fubini's theorem , no longer apply to non- finite measures (however, a transfer to more general cases is sometimes possible by applying the theorem to all -finite subspaces) . ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$
• The Birkhoff integral for Banach space -valent functions is defined with the aid of -finite measures. ${\displaystyle \sigma }$

Equivalence to probability measures

Two measures and on a common measuring space are called equivalent if they have the same zero quantities. That is, it is true as well as , they are mutually absolutely continuous . This actually explains an equivalence relation to measure. We further assume that it is not the zero measure. ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \mu \ll \nu }$ ${\displaystyle \nu \ll \mu }$ ${\displaystyle \mu \not \equiv 0}$

Many of the uses of finite measures now result from the following theorem: ${\displaystyle \sigma }$

Every -finite measure is equivalent to a probability measure . ${\displaystyle \sigma }$ ${\displaystyle \mu }$ ${\displaystyle P}$

The meaning of the proposition lies in the equivalence to a finite measure, even if is infinite. In particular, there is always an integrable function , so that applies to all . ${\displaystyle \mu (X)=\infty }$ ${\displaystyle \mu }$ ${\displaystyle w\in L^{1}(\mu )}$ ${\displaystyle 0<w(x)<1}$ ${\displaystyle x\in X}$

Definition for set functions

definition

A system of sets is given on the basic set , that is . Be ${\displaystyle {\mathcal {M}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {M}}\subset {\mathcal {P}}(X)}$

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

a positive set function. Then the set function is called -finite if there is a countable sequence of sets such that ${\displaystyle \sigma }$ ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle \bigcup _{n=1}^{\infty }A_{n}=X}$

applies and

${\displaystyle \mu (A_{n})<\infty {\text{ für alle }}n\in \mathbb {N} }$

applies. In particular, the quantity does not have to be included in the quantity system . ${\displaystyle X}$ ${\displaystyle {\mathcal {M}}}$

comment

With the above definition the -finitude can be extended to more general set functions. One of the most important uses of this term is Carathéodory's measure expansion theorem , according to which every -finite premeasure on a half-ring can be uniquely continued to a measure on the generated -algebra . Without the -finitude, the uniqueness does not follow here. ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Related terms

A term related to -finite measure is that of moderate measure . This is a Borel measure for which there is a countable covering of the basic set with open sets of finite measure. ${\displaystyle \sigma }$

There is also a concept of s-finiteness . A measure is called -finite if it is the countable sum of finite measures. Every -finite measure is always -finite, but not every -finite measure is -finite . ${\displaystyle \mu }$ ${\displaystyle s}$ ${\displaystyle \sigma }$ ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle \sigma }$

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 .
• Walter Rudin : Real and Complex Analysis . 3. Edition. McGraw-Hill, New York 1987 (English).