Tight measure

A tight measure is a mathematical term from measure theory , a branch of mathematics that deals with the investigation of abstract volume concepts and provides the basis for stochastics and integration theory . Tightness is a property that finite measures as well as families and sequences of finite measures can have. Rigid families of measures are used, for example, in the formulation of Prokhorov's theorem , where they are used to characterize weakly relatively compact sets of finite measures on Polish spaces . The weakly relatively sequence-compact sets are of great importance, since every sequence of elements from such a set always has a weak convergent subsequence .

definition

A metric space is given , provided with Borel's σ-algebra . ${\ displaystyle (X, d)}$ ${\ displaystyle {\ mathcal {B}} (X)}$

A finite measure on is called a tight measure if for each there is a compact amount such that ${\ displaystyle {\ mathcal {B}} (X)}$ ${\ displaystyle \ varepsilon> 0}$ ${\ displaystyle K \ subset X}$

{\ displaystyle \ mu (X \ setminus K) <\ varepsilon}

is. A set or family of finite measures is called rigid if for each there is a compact set such that ${\ displaystyle {\ mathcal {M}}}$ ${\ displaystyle \ varepsilon> 0}$ ${\ displaystyle K \ subset X}$

{\ displaystyle \ sup _ {\ mu \ in {\ mathcal {M}}} \; \ {\ mu (X \ setminus K) \} <\ varepsilon}

is. A sequence of finite measures is called tight if the set is tight. ${\ displaystyle (\ mu _ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle {\ mathcal {M}} = \ {\ mu _ {n} \; | \; n \ in \ mathbb {N} \}}$

For the special case of a probability measure it follows that it is tight if and only if there is a compact set for each such that ${\ displaystyle P}$ ${\ displaystyle P}$ ${\ displaystyle \ epsilon> 0}$ ${\ displaystyle K}$

{\ displaystyle P (K) \ geq 1- \ epsilon}

is. The tightness of sets, families and sequences of probability measures then follows analogously.

Examples

If the Dirac measure is on the point , understood as a measure , the sequence is not tight. Because the compact subsets of are limited and closed according to the Heine-Borel theorem . Then for each and every compact set there exists a , so that for all , there is limited. But this also applies to any compact quantity. So the sequence is not tight. ${\ displaystyle \ delta _ {x}}$ ${\ displaystyle x}$ ${\ displaystyle (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}$ ${\ displaystyle (\ delta _ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ varepsilon \ in (0,1)}$ ${\ displaystyle K}$ ${\ displaystyle N_ {K} \ in \ mathbb {N}}$ ${\ displaystyle N_ {K}> x}$ ${\ displaystyle x \ in K}$ ${\ displaystyle K}$ ${\ displaystyle \ delta _ {N_ {K}} (\ mathbb {R} \ setminus K) = 1}$

Conversely, the sequence is tight if and only if the sequence is restricted. Because if you put it in , the set is compact, and it is ${\ displaystyle {\ mathcal {M}}: = \ left (\ delta _ {a_ {n}} \ right) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle (a_ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle C: = \ sup _ {n \ in \ mathbb {N}} | a_ {n} |}$ ${\ displaystyle K = [- C, C]}$

{\ displaystyle \ sup _ {\ mu \ in {\ mathcal {M}}} \; \ {\ mu (\ mathbb {R} \ setminus [-C, C]) \} = 0}

and thus the tightness criterion is also fulfilled for everyone . ${\ displaystyle \ varepsilon> 0}$

comment

The term tightness is not clearly used in the literature, especially in the Anglo-Saxon language area. Elstrodt said in his German-language book of firmness and refers to the English term "tight", the Encyclopedia of Mathematics but directs tight measure on a locally finite measure on a Hausdorff space and the corresponding Borel σ-algebra , which regularly from the inside is . At Elstrodt, such dimensions are referred to as radon dimensions . The staffness also does not correspond to the English term tightness , this is regularity from within. It is therefore essential for every author to review the definitions used.

It is also sufficient if the room is only equipped with a topology and no metrics.

Related terms

The tightness can also be defined for distribution functions in the sense of stochastics , one then speaks of tight families of distribution functions .

literature

Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 380-400 , 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. 265-275 , doi : 10.1007 / 978-3-642-36018-3 .
Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 296 , doi : 10.1007 / 978-3-642-45387-8 .
Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 400-404 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

↑ Elstrodt: Measure and Integration Theory. 2009, p. 380.
↑ Tight measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
^ RA Minlos: Radon Mesure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

[1] Elstrodt: Measure and Integration Theory. 2009, p. 380.

[2] Tight measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

[3] RA Minlos: Radon Mesure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).