A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA M ATH EM ATIC A 7, 1995
Aleksander Kharazishvili
S O M E R E M A R K S O N N O N M E A S U R A B L E A L M O S T I N V A R I A N T S E T S
We discuss som e p ro p erties of alm ost invariant sets in conn ectio n w ith a m easure exten sio n problem . We consider a q uestion on th e ex isten ce of an alm o st inv arian t su b se t of a basic space, no nm ea- su ra b le w ith re sp ect to a given nonzero (r-fmite q u asiinv a ria n t (in p a rtic u la r, in v ariant) m easu re defined on th is space.
Let £ be a basic set and let G be a group of transform ations of E . In such situa tio n we say th a t the p air (E , G) is a space equipped w ith a tran sform a tio n group. Suppose also th a t fi is a cr-finite m easure defined on a <r-algebra of subsets of E . We say th a t a subset X of E is alm ost G -invariant w ith respect to the m easure fj, if, for every tra n sform ation g from G, we have the equality
K g ( X ) A X ) =
o
w here th e sym bol A denotes the operation of the sym m etric difference of sets.
Notice th a t alm ost invariant sets play an im p orta n t role in the gen-eral ergodic theory and, in particu lar, in some questions concerning extensions of quasiinvariant (respectively, invariant) m easures (see, for exam ple, [1], [2] or [3]). For instance, the following auxiliary
prop osition shows us th a t any cr-finite G -quasiinvariant (respectively, G -invariant) m easure defined on th e basic space E can be exten ded on to any alm ost G -invariant subset of E.
L e m m a 1. Let fi be a a-finite G-quasiinvariant (respectively, G- invariant) m easure defined on a a-algebra o f subsets o f E and let X be an alm ost G-invariant set w ith respect to fi. T hen there exists a m easure u defined on som e a-algebra o f subsets o f E such th at
1) v is a G -quasiinvariant (respectively, G -invariant) m easure; 2) v exten ds fi;
3) X is a v-m easurable set, i.e. X € d om (u ) .
T h e proof of Lem m a 1 is not difficult (see, for exam ple, [2] or [3]). From this lem m a we can conclude th a t, if a given alm ost invariant set X is nonm easurable w ith respect to the original m easure /i, then th e m easure v strictly extends fi. So we see th a t th e original nonzero cr-finite G -quasiinvariant (respectively, G -invariant) m easure fi can be stric tly extended provided th a t there exists a t least one subset of the space E not belonging to dom (p ) and alm ost G -invariant w ith respect to fi. Therefore, th e following question arises in a n a tu ra l way.
Q u e s t i o n 1. Let (E , G ) be a basic space equipped w ith a tra n sm a tio n group. W h at are the necessary and sufficient conditions, for-m ula ted in terfor-m s of the pair ( E , G), under which for every nonzero cr-finite G -quasiinvariant (respectively, G -invariant) m easure fi de-fined on E the re exists an alm ost G -invariant set w ith respect to ¡i not belonging to dom (/i)?
A nother, m ore interesting version of Q uestion 1 is th e following Q u e s t io n 2. Let again (E , G) be a basic space equipped w ith a tran sform a tio n group. W hat are the necessary and sufficient condi-tions, also form ulated in term s of the pair (E , G), und er which the re exists a countable fam ily {X n : n G u ) of subsets of E such th a t , for every nonzero a-finite G'-quasiinvariant (respectively, G -invariant) m easure fi defined on E , a t least one set X n is alm ost G -invariant w ith respect to fx and does not belong to dom( f i )?
R E M A R K S ON N O N M EA S U R A B L E S E T S 43
These tw o questions are still open. In the present pa p er we con-c en tra te o u r a tte n tio n on Q uestion 2. Namely, in the fu rth e r con- consid-era tion s we will show th a t in some pa rticu la r b ut im p o rta nt cases of ( E, G ) there exists a countable family of subsets of E w hich all are, in a ce rtain sense, alm ost G -invariant w ith respect to each nonzero (7-finite G -quasiinvariant (G -invariant) m easure ¡jl defined on E an d a t least one of these subsets is nonm easurable w ith respect to fi. We w ant to note here th a t the m ethod used in the furth e r considerations is ta ken from th e work [4] (see also [2] and [3]).
We need one auxiliary notion from the topological m easure theory. Let T be a topological space such th a t all one-elem ent subsets of T are Borel sets in T. We say th a t T is a Luzin space if every cr-finite diffused Borel m easure defined on T is identically equal to zero. Such spaces T are also called universally m easure zero topological spaces. Notice th a t the re are m any interesting exam ples of uncountab le un i-versally m easure zero subspaces of the real line R . O ne of the earliest exam ples is due to Luzin. Namely, using the m ethods of the theory of analytic sets, Luzin constructed a subset Z of the real line satisfying th e next tw o relations:
(1) c ard( Z) = u)\ w here u;j denotes the first unc ountable cardi-nal num ber;
(2) Z is a universally m easure zero space w ith respect to the induced topology.
T he con struction of th e m entioned set Z is given in detail in th e well-known m onograph of K uratowski [5]. From th e existence of th e set Z we im m ediately ob tain the following
L e m m a 2. L et Y be an arbitrary set o f cardinality u \ . T hen there exists a a-algebra S o f subsets o f Y satisfying the n e x t conditions:
1) all one-elem ent subsets o f Y belong to S; 2) S is a countably generated cr-algebra;
3) every a-finite diffused m easure defined on S is identically equal to zero.
L e m m a 3. Let E be a basic space w ith card( E) = uj\ and let G be a tra nsitive group o f transform ations o f E w ith card(G ) = u>i. T hen there exists a partition
{ Eq : a < u;i}
o f the space E such that
1) for each ordinal a < u>i th e set E a is at m o st countable; 2) for every subset A o f u \ and for every transform ation g from
the group G the set
(<7(U { E a : a € A }))A (U { £„ : a £ A}) is at m o st countable, as well.
T h e proof of Lem m a 3 is not difficult (see [2] or [3]). Notice th a t relation 2) of this Lem m a shows us, in particular, th a t for every subset A of U! th e corresponding set U {£ Q : a € i } is alm ost G -invariant w ith respect to any er-finite diffused G -quasiinvariant (G -invariant) m easure /i defined on the basic space E .
Taking into account the results of Lem m as 2 and 3, we o b ta in th e following
P r o p o s i t i o n 1. L et again E be a basic space o f cardinality u \ and let G be a transitive group o f transform ations o f E w ith the sam e cardinality. T hen there exists a countable fam ily { X n : n £ u } o f subsets o f E such that
1) for every a-finite diffused G-quasiinvariant (G -invariant) m easure ¡i defined on E each set X n (n € u ) is alm ost G-invariant w ith respect to fi;
2) for every nonzero a-finite G-quasiinvariant (G -invariant) m easure ¡i defined on E at least one set X n is nonm easurable w ith respect to ¡.l; moreover, there is an infinite n um be r o f sets from the fam ily { X n : n £ u>} which are nonm easurable w ith respect to [i.
R E M A R K S O N N O N M E A S U R A B LE S E T S 45
Proof. Indeed, applying the result of Lem m a 2, let us take a co u nt-able fam ily { A n : n € w} of subsets of u \ satisfying th e following conditions:
(1) <7-algebra S generated by this family contains all one-elem ent subsets of u \ ;
(2) any cr-fmite diffused m easure defined on S is identically equal to zero.
Now, let us pu t
X n = U {£ a : a € ¿ „ } ( n £ u ) .
T he n it is easy to check th a t the family of sets { X n : n 6 w} is a required one. We see also th a t for every nonzero <7-finite diffused m easure f.i defined on E at least one set X n is nonm easurable w ith respect to fi. and, m oreover, there is an infinite num ber of sets from th e fam ily { X n : n £ u } which are nonm easurable w ith respect to f.i.
Now, let us consider the case when ( E , G ) = ( R , R ) . In oth e r words, let us take the real line R equipped w ith the group of all its tran slatio ns (of course, we can identify the additive group of R w ith th e group of all translatio ns of R by the canonical isom orphism betw een these two groups).
Using a Ham el base of R , we can represent R as a direct sum R = G i + G‘2 (G i n (?2 = {0})
of tw o subgroups G i and G2 in such a way th a t c ard( Gi ) = ujy. D enote by th e sym bol I the ideal of subsets of R generated by the fam ily
[ Y + G2 : Y is a countable subset of G\ }. It is easy to check th a t
(1) I is a er-ideal of subsets of R;
(2) I is invariant under all translation s of R;
(3) for each set Z £ I there exists an uncountable fam ily {ga : a < u \ } of elem ents of the group G\ such th a t
{ga + Z : a < u>i} is a pairw ise disjoint family of sets.
F rom re lation (3) we can conclude th a t for every <7-finite R -quasi- in variant (respectively, R -invariant) m easure // defined on R the equality
M ^) = o
( z e I)holds (the sym bol //* denotes here the inner m easure associated w ith fi).
Therefore, we have the following
L e m m a 4. L et \i be any a-finite R-quasiinvariant (respectively, R - invariant) m easure defined on R. T hen there exists a m easure v defined on R and satisfying the ne xt relations:
1) v is an R-quasiinvariant (respectively, R -invariant) m easure; 2 ) v e xtends n;
3) I C dom (u);
4) v ( Z ) = 0 for each set Z £ I.
Now, let us consider the group G\ as a basic space equipped w ith th e group of transform ations G \ . Since card(G \ ) = u>\ a nd the group G] acts tra nsitively on G i, we can apply to G\ the result of P roposi-tion 1. According to this proposition there exists a countable fam ily {Y n : n £ u;} of subsets of G\ such th a t
(a) for each index n £ w and for each tran slatio n g £ G \ we have card((g + F „ )A F „ ) < u>]
(b) for every nonzero <7-finite G i-quasiinvai’iant (respectively, G \- inv ariant) m easure ¡j, defined on G\ a t least one set F„ is nonm easur- able w ith respect to //; moreover, there is an infinite num ber of sets from th e fam ily {Fn : n £ a;} which are nonm easurable w ith respect to
Let us p u t
= Fn + G2 (n £ u;).
T he n it is obvious th a t for each index n £ uj and for each tran sla tion h £ R we have
R E M A R K S O N N O N M E A SU R A B LE S E T S 47
O f course, in th is case we can not assert th a t th e sets X n(n E u ) are alm ost R -invariant in the basic space R. Indeed, if I is the classical Lebesgue m easure on R and the group G2 is not a Lebesgue m easure zero set in R , then there exists a set X n which is not alm ost R- inv ariant w ith respect to / (m oreover, there is an infinite num ber of sets from the family {X n : n £ u>] which are not alm ost R -invariant w ith respect to /). But , taking into account th e prope rties of the ideal I m entioned above, we see th a t all sets X n (n £ u j) are alm ost R -invariant w ith respect to a certain m easure I' w hich is defined on R , extends I and is also an R -invariant m easure.
More generally, after the preceding rem arks it is clear th a t we have th e following result.
P r o p o s i t i o n 2. T h e countable fam ily o f sets {X„ : n 6 u} m e n -tioned above satisfies the n ex t relations:
1) for every cr-finite R -quasiinvai'iant (R-invariant) m easure /i defined on R there exists an R -quasiinvariant (R -invariant) m easure v defined on R and extending ¡-i such th at all sets X n (n 6 u ) a re alm ost R -invariant w ith respect to u;
2) for every nonzero a-finite R-quasiinvai'iant (R -invariant) m easure /i defined on R there exists at least one set X n non- m easurable w ith respect to /j; moreover, there is an infinite n um be r o f sets from the fam ily {X„ : n 6 u ] w hich are nonm easurable w ith respect to ¡.i.
Rem ark 1. It is not difficult to see th a t a result analogous to P ro po -sition 2 is valid for th e m -dim ensional Euclidean space R m equipped w ith th e group of all translation s of R m and, m ore generally, for an a rb itra ry uncountable vector space E equipped w ith th e group of all tra nslation s of E . It can be also shown th a t the sam e result is tru e for some classes of uncountable groups equipped w ith th e groups of all th e ir left tra nslations. B ut the following question is still open: let T be any uncountable group equipped with the group of all its left tran slation s; is it tru e an analogue of Proposition 2 for T?
Rem ark 2. Let E be a basic space and let G be a group of tra n s-form ations of E . Suppose th a t is a cr-finite G '-quasiinvariant
(G in varian t) com plete m easure defined on a eralgebra S(f i ) of su b -sets of E. Let X be an a rb itra ry subset of E nonm easurable and alm ost G -invariant w ith respect to /i. Then, according to Lem m a 1, the re exists a com plete m easure v defined on a cr-algebra S ( v ) of subsets of E and satisfying the following conditions:
1) v is a G-quasiinvariant, (G -invariant) m easure; 2) i/ strictly extends the original m easure /<; 3) th e set X belongs to S(u).
Let us p u t
/ ( / /) = the ideal of all /¿-measure zero sets; I ( v ) = th e ideal of all I'-measure zero sets.
It is clear th a t we have two m easure B oolean algebras A M = S W / I M , A( u) = S M / I M and, since v extends //, we have the canonical em bedding
<f>: A( n) -> A(u).
It is not difficult to check th a t the m easure v can be taken in such a way th a t th e m apping <j) will not be a surjection. Hence, for this m ea-sure v, th e corresponding m eaea-sure a lgebra A( u) contains the m eaea-sure algebra A(f i ) as a proper subalgebra.
Finally, let us form ulate th e th ird proposition concerning th e exis-tence of nonm easurable alm ost invariant subsets of a basic space E. P r o p o s i t i o n 3 . L et E be a n o n e m p ty basic set and Jet G be a group o f transform ations o f E w ith card(G) = ui\, acting freely on E (in particular, from this condition it follows that E is an uncountable space). T hen there exists a countable fam ily { X n : n 6 u>} o f subsets o f E such that
1) for every nonzero a-finite G-quasiinvariant (G -invariant) m easure /j defined on E at least one set X n is nonm easurable
R E M A R K S ON N O N M E A S U R A B L E S E T S 49
w ith respect to n; moreover, there is an infinite nu m ber o f sets from this fa m ily which are nonm easurable w ith respect
to
2) for every o-finite G-quasiinvariant (G -invariant) m easure ¡i defined on E there exists a G-quasiinvariant (G -invariant) m easure u defined on E exte nding /x and having the pro p erty th at all sets X n (n 6 u j) are alm ost G-invariant w ith respect to v.
T he proof of this proposition is sim ilar (in some details) to the pro of of P rop ositio n 1.
Rem ark 3. Let (E , G) be a basic space equipped w ith a tra nsfo rm a -tion group. It is not difficult to check th a t the following tw o sentences are equivalent:
1) all G -orbits in E are uncountable;
2) every ¿r-finite G -quasiinvariant (respectively, G -invariant) m easure defined on E can be extended to a cr-finite diffused G -quasiinvariant (respectively, G -invariant) m easure defined on E .
In p a rticu la r, if E is an uncountable basic space and a group G of tran sfo rm a tio ns of E acts transitively on E , then every ¿r-finite G- quasiinvariant (respectively, G -invariant) m easure defined on E can be ex tended to a a -finite diffused G -quasiinvariant (respectively, G- inv a ria nt) m easure defined on E .
Re f e r e n c e s
[1] S. K a k u tan i, J . O x to by , C o n stru ction o f a nonseparable in va ria n t exten sio n
o f the Lebesgue m easure space, A nn . M ath . 52 (1950), 580-590.
[2] A .B . K harazish vili, So m e questions o f S et theory and M easure theory, (in R u ssian ), Izd. T b il. Gos. U niv. T b ilisi (1978).
[3] A .B . K h arazish vili, In varia nt extensions o f the Lebesgue m easure, (in R us-sia n), Izd. T b il. Gos. U niv. T bilisi (1983).
[4] A .B . K harazishvili, S o m e applications o f H am el bases, (in R ussia n), B ull. A cad. G eorgian SS R 85 no. 1 (1977), 17-20.
Aleksander Kharazishvili
P E W N E U W A G I O P R A W I E N I E Z M I E N N I C Z Y C H Z B I O R A C H N I E M I E R Z A L N Y C H
W pracy rozw aża się pew ne własności praw ie niezm ienniczych zbio-rów w kontekście problem u przedłużania m iar niezm ienniczych.
In stitu te of A pplied M a th e m atics U n iversity of T b ilisi U niversity S tr. 2, 380043 T b ilisi 43, G eo rgia
