R = {(x, Tgx)\x X, g G G} is a nonsingular equivalence relation, and
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
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
such that, for every
F G [ R ] ,
(dß/dßV)(x) = PRtß(x, Vx) for ì-a.e .
÷ e X. The map ì
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)
(1.2) ì*\Â)= t\{xeX:(x,x)eRnB}\dß{x)
for every Borel set  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
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
), 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
_ 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 à Ã/ =
à / ì-a.e. , for all g, g G G. The action à is measure preserving if ì is invariant under
every Ô' g G G, and à is ergodic if every  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
S^', ì' ) is conjugate to à 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)
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 ÷
and Ã' (as well as Ô ) is called transitive if there exists a point
x e l ' with ì{×\Ô'0÷) = 0 . The actions à and
are properly ergodic if they are ergodic
and not transitive.
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