Vague Convergence (Measure Theory)

The vague convergence is a Konvergenzart in the measure theory , a branch of mathematics which deals with abstracted volume terms and the basis for the stochastic and the integration theory forms. The vague convergence is a convergence term for sequences of radon measures and differs from the weak convergence by this and by the choice of a different class of test functions . The topology that describes the vague convergence is called the vague topology .

definition

A locally compact Hausdorff space is given and the corresponding Borel σ-algebra is given . Also, be Radon measures on , that is, each of these dimensions is ${\displaystyle (X,d)}$ ${\displaystyle {\mathcal {B}}={\mathcal {B}}(X)}$ ${\displaystyle (\mu _{n})_{n\in \mathbb {N} },\mu }$ ${\displaystyle {\mathcal {B}}}$

1. locally finite, that is, for each there is an open environment of finite measure, ${\displaystyle x\in X}$ ${\displaystyle x}$
2. regular from the inside .

The sequence of measures is then called vaguely convergent to the measure if for every continuous function with a compact support ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu }$ ${\displaystyle f\colon X\to \mathbb {K} }$

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

applies. It then writes vague, or . ${\displaystyle \mu _{n}\rightarrow \mu }$ ${\displaystyle \mu _{n}{\xrightarrow[{}]{v}}\mu }$ ${\displaystyle \mu =v{\text{-}}\!\lim _{n\to \infty }\mu _{n}}$

comment

With the definition, caution is advised in two places: First, the term radon measure is not used clearly in the literature and should therefore always be compared. Second, with the convergence of dimensions, a fine gradation of the convergence terms is possible, which is characterized by a different choice of test functions. You should therefore always pay attention to which class of test functions is used in order to avoid possible errors.

Motivation to define

Intuitively one would say of a sequence of measures that it converges to if ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu }$

${\displaystyle \lim _{n\to \infty }\mu _{n}(A)=\mu (A)}$

holds for every set from the considered σ-algebra. But if you now use the measuring room as a sequence of dimensions, for example ${\displaystyle A}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

${\displaystyle \mu _{n}(A)=\delta _{\tfrac {1}{n}}(A)}$

the Dirac measures in each case in the point , one would "intuitively" expect that the sequence converges to the Dirac measure in the point . But this is not the case, as you can see from the crowd , for example , because it is ${\displaystyle {\tfrac {1}{n}}}$ ${\displaystyle \delta _{0}}$ ${\displaystyle 0}$ ${\displaystyle A=(-\infty ,0]}$

${\displaystyle \lim _{n\to \infty }\delta _{\tfrac {1}{n}}(A)=0\neq \delta _{0}(A)=1}$-

So the concept of convergence is too strong. An equivalent formulation of the above, intuitive concept of convergence for sequences of measures is

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

for all , i.e. the essentially limited functions . On the basis of this characterization, one now looks for weaker functional classes and sets of measures , so that the above equation still applies to this choice and is also a separating family for . So it should also ${\displaystyle f\in {\mathcal {L}}^{\infty }(X,{\mathcal {B}}(X),\mu )}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle \int _{X}f\mathrm {d} \mu =\int _{X}f\mathrm {d} \nu {\text{ für alle }}f\in {\mathcal {F}}\implies \mu =\nu }$

be valid. This guarantees the uniqueness of the limit value. If one now chooses as the Radon measures on the Borel σ-algebra of a locally compact Hausdorff space and as the continuous functions on a compact support, one obtains the vague convergence described here. Another choice of the function classes and sets of measures provides, for example, the weak convergence in the sense of measure theory or the convergence in the distribution of stochastics. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {F}}}$

properties

• According to Helly-Bray's theorem , measures converge to vague if and only if the associated distribution functions converge vaguely to a distribution function except for constants . ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$
• After the selection of Helly's theorem every bounded sequence has of measures on a vague convergent subsequence . A sequence of measures is called restricted if the sequence of the norms of total variation is restricted. ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle (\|\mu _{n}\|)_{n\in \mathbb {N} }}$
• It can be shown that if is locally compact and Polish , the following two statements are equivalent: ${\displaystyle (X,d)}$

1. ${\displaystyle \mu =v{\text{-}}\!\lim _{n\to \infty }\mu _{n}{\text{ und }}\mu (X)=\lim _{n\to \infty }\mu _{n}(X)}$
2. ${\displaystyle \mu =v{\text{-}}\!\lim _{n\to \infty }\mu _{n}{\text{ und }}\mu (X)\geq \limsup _{n\to \infty }\mu _{n}(X)}$.

This is also occasionally formulated as an addition to the Portmanteau theorem .

Vague topology

The vague convergence can be described by a topology, the so-called vague topology . It is the coarsest topology, so that all mappings

${\displaystyle \mu \mapsto \int f\mathrm {d} \mu }$

are continuous for all continuous functions with compact support.

Web links

• Vague topology . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 380-392 , 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 , p. 256-265 , doi : 10.1007 / 978-3-642-36018-3 .