# 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)

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.