Quasi-invariant measure

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.