Namespaces
Variants
Actions

Difference between revisions of "Quasi-invariant measure"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A measure on a space that is equivalent to itself under  "translations"  of this space. More precisely: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765601.png" /> be a [[Measurable space|measurable space]] (that is, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765602.png" /> with a distinguished <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765603.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765604.png" /> of subsets of it) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765605.png" /> be a group of automorphisms of it (that is, one-to-one transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765606.png" /> that are measurable together with their inverses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765607.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765608.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q0765609.png" />). A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656011.png" /> is said to be quasi-invariant (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656012.png" />) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656013.png" /> the transformed measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656015.png" />, is equivalent to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656016.png" /> (that is, these measures are absolutely continuous with respect to each other, cf. [[Absolute continuity|Absolute continuity]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656017.png" /> is a topological [[Homogeneous space|homogeneous space]] with a continuous locally compact group of automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656018.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656019.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656020.png" /> and is endowed with a topology such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656022.png" />, is continuous with respect to the product topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656023.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656024.png" /> is the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656025.png" />-algebra with respect to the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656026.png" />, then there exists a quasi-invariant measure that is unique up to equivalence [[#References|[1]]]. In particular, a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656027.png" /> is quasi-invariant with respect to all shifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656029.png" />, if and only if it is equivalent to [[Lebesgue measure|Lebesgue measure]]. If the group of transformations is not locally compact, there need not be a quasi-invariant measure; this is the case, for example, in a wide class of infinite-dimensional topological vector spaces [[#References|[2]]].
+
<!--
 +
q0765601.png
 +
$#A+1 = 49 n = 0
 +
$#C+1 = 49 : ~/encyclopedia/old_files/data/Q076/Q.0706560 Quasi\AAhinvariant measure
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A measure on a space that is equivalent to itself under  "translations"  of this space. More precisely: Let $  ( X , B ) $
 +
be a [[Measurable space|measurable space]] (that is, a set $  X $
 +
with a distinguished $  \sigma $-
 +
algebra $  B $
 +
of subsets of it) and let $  G $
 +
be a group of automorphisms of it (that is, one-to-one transformations $  g : X \rightarrow X $
 +
that are measurable together with their inverses $  g  ^ {-} 1 $
 +
with respect to the $  \sigma $-
 +
algebra $  B $).  
 +
A measure $  \mu $
 +
on $  ( X , B ) $
 +
is said to be quasi-invariant (with respect to $  G $)  
 +
if for any $  g \in G $
 +
the transformed measure $  g \mu ( A) = \mu ( g  ^ {-} 1 A ) $,  
 +
$  A \in B $,  
 +
is equivalent to the measure $  \mu $(
 +
that is, these measures are absolutely continuous with respect to each other, cf. [[Absolute continuity|Absolute continuity]]). If $  X $
 +
is a topological [[Homogeneous space|homogeneous space]] with a continuous locally compact group of automorphisms $  G $(
 +
that is, $  G $
 +
acts transitively on $  X $
 +
and is endowed with a topology such that the mapping $  G \times X \rightarrow X $,
 +
$  ( g , x ) \rightarrow g x $,  
 +
is continuous with respect to the product topology on $  G \times X $)  
 +
and $  B $
 +
is the Borel $  \sigma $-
 +
algebra with respect to the topology on $  X $,  
 +
then there exists a quasi-invariant measure that is unique up to equivalence [[#References|[1]]]. In particular, a measure on $  \mathbf R  ^ {n} $
 +
is quasi-invariant with respect to all shifts $  x \rightarrow x + a $,
 +
$  x , a \in \mathbf R  ^ {n} $,  
 +
if and only if it is equivalent to [[Lebesgue measure|Lebesgue measure]]. If the group of transformations is not locally compact, there need not be a quasi-invariant measure; this is the case, for example, in a wide class of infinite-dimensional topological vector spaces [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Thus, a quasi-invariant measure is a generalization of a [[Haar measure|Haar measure]] on a topological group. On a locally compact group with left Haar measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656030.png" /> a measure is left quasi-invariant (quasi-invariant under left translations) if and only if it is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656031.png" />.
+
Thus, a quasi-invariant measure is a generalization of a [[Haar measure|Haar measure]] on a topological group. On a locally compact group with left Haar measure $  \mu $
 +
a measure is left quasi-invariant (quasi-invariant under left translations) if and only if it is equivalent to $  \mu $.
  
There exists no quasi-invariant measure on an infinite-dimensional Hilbert space with respect to the group of all translations (and so, in particular, no Haar measure). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656032.png" /> be a rigged Hilbert space, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656033.png" /> a nuclear space with inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656035.png" /> the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656037.png" /> the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656038.png" />. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656039.png" /> defines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656041.png" />, the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656042.png" />. A measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656044.png" /> is quasi-invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656045.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656048.png" />, i.e. if it is quasi-invariant with respect to the group of translations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076560/q07656049.png" />. There exist quasi-invariant measures on such dual spaces of nuclear spaces, [[#References|[2]]], Chapt. IV, §5.2.
+
There exists no quasi-invariant measure on an infinite-dimensional Hilbert space with respect to the group of all translations (and so, in particular, no Haar measure). Let $  \Phi \subset  H \subset  \Phi  ^  \prime  $
 +
be a rigged Hilbert space, with $  \Phi $
 +
a nuclear space with inner product $  (  , ) $,  
 +
$  H $
 +
the completion of $  \Phi $,  
 +
and $  \Phi  ^  \prime  $
 +
the dual of $  \Phi $.  
 +
Each $  f \in \Phi $
 +
defines an element $  F _ {f} $
 +
in $  \Phi  ^  \prime  $,  
 +
the functional $  F _ {f} ( g) = \langle  f , g\rangle $.  
 +
A measure $  \mu $
 +
on $  \Phi  ^  \prime  $
 +
is quasi-invariant if $  \mu ( F _ {f} + X) = 0 $
 +
for all $  f \in \Phi $
 +
and $  X \subset  \Phi  ^  \prime  $
 +
with $  \mu ( X) = 0 $,  
 +
i.e. if it is quasi-invariant with respect to the group of translations $  \{ {F _ {f} } : {f \in \Phi } \} $.  
 +
There exist quasi-invariant measures on such dual spaces of nuclear spaces, [[#References|[2]]], Chapt. IV, §5.2.

Revision as of 08:09, 6 June 2020


A measure on a space that is equivalent to itself under "translations" of this space. More precisely: Let $ ( X , B ) $ be a measurable space (that is, a set $ X $ with a distinguished $ \sigma $- algebra $ B $ of subsets of it) and let $ G $ be a group of automorphisms of it (that is, one-to-one transformations $ g : X \rightarrow X $ that are measurable together with their inverses $ g ^ {-} 1 $ with respect to the $ \sigma $- algebra $ B $). A measure $ \mu $ on $ ( X , B ) $ is said to be quasi-invariant (with respect to $ G $) if for any $ g \in G $ the transformed measure $ g \mu ( A) = \mu ( g ^ {-} 1 A ) $, $ A \in B $, is equivalent to the measure $ \mu $( that is, these measures are absolutely continuous with respect to each other, cf. Absolute continuity). If $ X $ is a topological homogeneous space with a continuous locally compact group of automorphisms $ G $( that is, $ G $ acts transitively on $ X $ and is endowed with a topology such that the mapping $ G \times X \rightarrow X $, $ ( g , x ) \rightarrow g x $, is continuous with respect to the product topology on $ G \times X $) and $ B $ is the Borel $ \sigma $- algebra with respect to the topology on $ X $, then there exists a quasi-invariant measure that is unique up to equivalence [1]. In particular, a measure on $ \mathbf R ^ {n} $ is quasi-invariant with respect to all shifts $ x \rightarrow x + a $, $ x , a \in \mathbf R ^ {n} $, if and only if it is equivalent to Lebesgue measure. If the group of transformations is not locally compact, there need not be a quasi-invariant measure; this is the case, for example, in a wide class of infinite-dimensional topological vector spaces [2].

References

[1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[2] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1964) (Translated from Russian)

Comments

Thus, a quasi-invariant measure is a generalization of a Haar measure on a topological group. On a locally compact group with left Haar measure $ \mu $ a measure is left quasi-invariant (quasi-invariant under left translations) if and only if it is equivalent to $ \mu $.

There exists no quasi-invariant measure on an infinite-dimensional Hilbert space with respect to the group of all translations (and so, in particular, no Haar measure). Let $ \Phi \subset H \subset \Phi ^ \prime $ be a rigged Hilbert space, with $ \Phi $ a nuclear space with inner product $ ( , ) $, $ H $ the completion of $ \Phi $, and $ \Phi ^ \prime $ the dual of $ \Phi $. Each $ f \in \Phi $ defines an element $ F _ {f} $ in $ \Phi ^ \prime $, the functional $ F _ {f} ( g) = \langle f , g\rangle $. A measure $ \mu $ on $ \Phi ^ \prime $ is quasi-invariant if $ \mu ( F _ {f} + X) = 0 $ for all $ f \in \Phi $ and $ X \subset \Phi ^ \prime $ with $ \mu ( X) = 0 $, i.e. if it is quasi-invariant with respect to the group of translations $ \{ {F _ {f} } : {f \in \Phi } \} $. There exist quasi-invariant measures on such dual spaces of nuclear spaces, [2], Chapt. IV, §5.2.

How to Cite This Entry:
Quasi-invariant measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-invariant_measure&oldid=18836
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article