OPERATOR ALGEBRAS AND DYNAMICAL SYSTEMS

5

Ô

R = {(x, Tgx)\x € X, g G G} is a nonsingular equivalence relation, and

RT

is transitive or ergodic if and only if Ô is transitive or ergodic. Every

nonsingular equivalence relation R on (X, S?, ì ) is of the form R = R

r

for some nonsingular action Ô of a countable group on (X, S?, ì ) [FelMl].

We dehne the /w// group [R] of R as the group of all nonsingular automor-

phisms V of (X, S", ì) with (je, Vx) e R for ì-a.e . ÷ € X . Then

there exists a Borel map (x, x') - /?R (x, x') = Üì{÷)/Üì{÷) from R to

R+

such that, for every

F G [ R ] ,

(dß/dßV)(x) = PRtß(x, Vx) for ì-a.e .

÷ e X. The map ì

ê

is the Radon-Nikodym derivative of the equivalence

relation R. If /?R = 1 then R preserves ì (or ì is R-invariant). Consider

the ó-finite measures ì ^ and ì ^ on R defined by

(1.1) ìÀ\Â)= f\{xeX:(x,x)eRnB}\dß(x)

and

(1.2) ì*\Â)= t\{xeX:(x,x)eRnB}\dß{x)

for every Borel set Â c R, where |5| denotes the cardinality of a set S.

These measures are equivalent (i.e. have the same null sets), and

(1.3) ^ì^/Üì^ = ñÊì .

If we are only interested in the measure dass of ì ^ ' we write ì

ê

to denote

either ì^

}

or ì

{*].

(2) Induced equivalence relations. Let R be a Borel equivalence relation

on (X, S?), Â eS*, and let RB = Rn{B xB) be the equivalence relation

induced by R on 5 . If ì is a measure on X and ì(Â) 0 then R5 is

obviously a nonsingular equivalence relation on (Â, 5^, ì

â

), where ^ =

If Ê is invertible (up to a null set) then V is a nonsingular automorphism of (Ë", J^, ì) .

In both cases the measure ì is said to be quasi-invariant under V , and ì is invariant if

ìÊ

_ 1

= ì . Á nonsingular action Ô of a group G on (Á, ^ , ì ) is a map g -+ Tg from G

into the group Aut(A, &, ì) of nonsingular automorphisms of {X, S", ì ) suchthat Ã Ã/ =

Ã / ì-a.e. , for all g, g G G. The action Ã is measure preserving if ì is invariant under

every Ô' g G G, and Ã is ergodic if every Â e 5* with ì(2?Ä7^1? ) = 0 for all g e G

satisfies that either ì(2? ) = 0 or ì(×\Â) = 0. Á second nonsingular action T' of G on

a measure Space

(A;,

S^', ì' ) is conjugate to Ã if there exists a nonsingular isomorphem

^ : ( Á , ^ , ì ) ^ ( Á ' , ^ ' , ì ' ) such that, for every ^ G , ØÔ§ = Tgy/ ì-a.e .

If the group G is locally compact and second countable then every nonsingular action Ô

of G on (X, J^, ì ) is conjugate to a nonsingular action Ô of G on a measure Space

(X1, J^7',

ì' ) with the property that the map (g, x) —

T1

x from G x l ' to X' is Borel,

and Ô'Ô', = Ã' / for all ^ , / G G (cf. [Var]). In this case the orbit TGx = {Tgx: g G G} is

ï ï ï ï ï

a Borel set for every ÷ €

Xf,

and Ã' (as well as Ô ) is called transitive if there exists a point

x e l ' with ì{×\Ô'0÷) = 0 . The actions Ã and

T1

are properly ergodic if they are ergodic

and not transitive.