View source for Quasi-invariant measure

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]]].

====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>



====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" />.

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.