Finite measure

A finite measure is a term from measure theory , a branch of mathematics that deals with abstract volume concepts. A finite measure is clearly a concept of volume in which the basic set under consideration only has a finite volume. The best known example of finite measures are the probability measures in stochastics . These are exactly the finite measures for which the basic set has the volume 1. The volume is then interpreted as a probability in this case.

Despite the simplicity of the definition, finite measures have a variety of properties that depend on the structures (base set and σ-algebra ) on which they are defined.

definition

The following notations are agreed for the entire article:

• ${\displaystyle X}$ be an arbitrary set, the basic set
• ${\displaystyle {\mathcal {A}}}$be any σ-algebra on the basic set
• ${\displaystyle {\mathcal {B}}}$or denotes the Borel σ-algebra on when at least one topological space is. ${\displaystyle {\mathcal {B}}(X)}$ ${\displaystyle X}$ ${\displaystyle X}$

A measure on the measuring space is called a finite measure if is. ${\displaystyle \mu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \mu (X)<\infty }$

This means in full: A finite measure is a set function

${\displaystyle \mu :{\mathcal {A}}\to [0,\infty )}$

from a σ-algebra over the basic set into the non-negative real numbers with the following properties: ${\displaystyle X}$

• ${\displaystyle \mu (\emptyset )=0}$
• ${\displaystyle \mu (X)<\infty }$
• σ-additivity : For every sequence of pairwise disjoint sets from holds . ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n})}$

We denote with the set of finite measures on the base space and the σ-algebra . Different spellings can be found in the literature: In some cases, the σ-algebra is omitted ( or similar) if this is evident from the context, and in some cases the specification of the basic set such as, for example . Or there are other indices such as a subscript f, for the English “finite” ( finite ). The superscript plus is often found when rooms with signed dimensions are also used, the "usual" dimensions then correspond to the positive elements in this room. ${\displaystyle {\mathcal {M}}^{+}(X,{\mathcal {A}})}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {M}}(X)}$ ${\displaystyle {\mathcal {M}}^{+}({\mathcal {B}})}$ ${\displaystyle {\mathcal {M}}_{f}}$

Properties as a measure

The following properties infer directly from the fact that every finite measure is a measure.

• Subtractivity : For with and : ${\displaystyle A,B\in {\mathcal {A}}}$ ${\displaystyle B\subseteq A}$ ${\displaystyle \mu (A)<\infty }$

${\displaystyle \mu (A\setminus B)=\mu (A)-\mu (B)}$.

• Monotony : A finite measure is a monotonous mapping from to , that is, applies to ${\displaystyle ({\mathcal {A}},\subset )}$ ${\displaystyle ([0,\infty ),\leq )}$ ${\displaystyle A,B\in {\mathcal {A}}}$

${\displaystyle B\subset A\implies \mu (B)\leq \mu (A)}$.

• Finite additivity : From the σ-additivity it follows directly that it holds for pairwise disjoint sets ${\displaystyle A_{1},\dotsc ,A_{m}\in {\mathcal {A}}}$

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

• σ-subadditivity : For any sequenceof sets intrue ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$

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

• σ-continuity from below : Isa monotonic against increasing set sequence in, so, so is. ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle A_{n}\uparrow A}$ ${\displaystyle \lim _{n\to \infty }\mu (A_{n})=\mu (A)}$
• σ-continuity from above : Ifa monotonic against decreasing set sequence in, so, then is. ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle A_{n}\downarrow A}$ ${\displaystyle \lim _{n\to \infty }\mu (A_{n})=\mu (A)}$
• Principle of inclusion and exclusion : It applies

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

such as

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

In the simplest case this corresponds to

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

Properties on different base quantities

For any arbitrary, but firmly selected measurement space , the finite dimensions are a subset of the real vector space of the finite, signed dimensions in this measurement space. In this vector space they form a convex cone . ${\displaystyle (X,{\mathcal {A}})}$

Important convex subsets of the finite measures are the probability measures (those elements with ) and the sub-probability measures (those elements with ). ${\displaystyle \mu (X)=1}$ ${\displaystyle \mu (X)\leq 1}$

As a subset of the finite signed measures, the total variation norm for finite measures is defined as

${\displaystyle \|\mu \|_{\operatorname {TV} }=\mu (X)}$

and enables a concept of convergence.

On topological spaces

Is a Hausdorff space and contains the Borel σ-algebra , then each to immediately locally finite measure . So every finite measure is automatically a Borel measure . ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$

Each finite regular inside dimension to (read: each finite Radon measure ) is a regular level , because then the regularity of the inside of the amount from the outside of the regularity of the amount corresponds. ${\displaystyle {\mathcal {B}}}$ ${\displaystyle A}$ ${\displaystyle X\setminus A}$

On metric spaces

If there is a metric space , then the weak convergence can be defined for finite measures : A sequence of finite measures is called weakly convergent to , if ${\displaystyle X}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu }$

${\displaystyle \lim _{n\to \infty }\int _{X}f\mathrm {d} \mu _{n}=\int _{X}f\mathrm {d} \mu }$

holds for all bounded continuous functions . The Portmanteau theorem provides further characterizations of the weak convergence . ${\displaystyle f}$

The Prokhorov metric defines a metric in terms of finite measures and thus turns it into a metric space that is separable if and only if it is separable. ${\displaystyle d_{P}}$ ${\displaystyle ({\mathcal {M}}^{+}(X,{\mathcal {B}}),d_{P})}$ ${\displaystyle X}$

For separable fundamental sets, a sequence of measures converges weakly if and only if it converges with respect to the Prokhorov metric. The Prokhorov metric thus measures the weak convergence.

Furthermore, Prokhorov's theorem characterizes the relatively sequentially compact sets (with regard to weak convergence): If a set of finite measures is tight and bounded, then it is relatively sequentially compact.

On Polish rooms

If there is a Polish space , then by Ulam's theorem, every finite measure is a regular measure. ${\displaystyle X}$ ${\displaystyle {\mathcal {B}}}$

The properties of the basic space are inherited by the set of measures: is Polish if and only if is Polish. ${\displaystyle ({\mathcal {M}}^{+}(X,{\mathcal {B}}),d_{P})}$ ${\displaystyle X}$

In addition, Prokhorov's theorem provides a stronger characterization of weakly relatively sequentially compact sets: A set of measures is weakly relatively sequentially compact if and only if it is tight and restricted.

Generalizations

σ-finite measures

→ Main article : σ-finite measure

σ-finite measures try to get some of the properties of a finite measure by demanding that the basic set can be divided into countably many sets of finite measure. Thus σ-finite measures are not “too big”. A measure in a measuring space is called σ-finite if there are sets such that ${\displaystyle \mu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle A_{n}\in {\mathcal {A}}}$

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

and for everyone ${\displaystyle \mu (A_{n})<\infty }$ ${\displaystyle n\in \mathbb {N} }$

Moderate dimensions

→ Main article : Moderate measure

Moderate measures are a tightening of σ-finite measures and serve to derive regularity criteria from non-finite Borel measures. A Borel measure is called a moderate measure if there are countably many open sets with ${\displaystyle (O_{n})_{n\in \mathbb {N} }}$

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

and ${\displaystyle \mu (O_{n})<\infty }$

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 .