A (' T A U N 1 V E R S 1 T A T I S L 0 D Z 1 E N S I S FOLIA M AT HE M ATICA 9, 1997
/1 leksan d er h ’h arazishvili
ON VITALI CONSTRUCTION
FOR COMMUTATIVE GROUPS
WITH QUASIINVARIANT MEASURES
W e d is c u s s an a n a lo g u e o f t h e clas sical V it a l i c o n s t r u c t io n o f a L e b e s g u e n o n m e a s u r a b l e set. for u n c o u n t a b le c o m m u t a t i v e g r o u p s e q u i p p e d w ith n on z er o ir - f ini te q u a s iin v ar ia n t m e as u r e s .
Let R be the ad ditiv e group of the real line, Q be the subgroup of R co nsisting of all rational num bers and let A deno te the standard L ebesgue m easure on R. T he classical V itali theorem (see [13]) states tha t every selector o f the fam ily R / Q is nonm easurable with respect to A (th e sam e result holds for any counta ble dense subgroup H o f R and for all selectors o f the fam ily R / I I) . T his im portant theorem was generalized in several directions (see, e.g .. [l], [2], [5], [8]. [11], [12], [14]). In particular, som e generalizations and analogues of the V itali theorem were obtained for locally co m pact topological groups equipped w ith th e Haar m easure (see, for instance, [4] or [3], Section 4). T he aim o f the present paper is to show tha t m any questions arising na t-urally in connection with this theorem can com pletely be solved, for uncountable com m uta tive groups, w ithout the aid o f any topological m etho ds. Nam ely, we shall establish, in our further considerations, the corresponding analogue o f the V itali theorem for uncountable com m u-ta tiv e groups equipped with nonzero a - fin ite quasiinvariant m easures.
T he notation and term inology used in the present paper are stan-dard. As usual, u> denotes t ho first infinite cardinal num ber, denotes the first uncountable cardinal num ber, denotes the cardinality o f the co ntinuum . A sot ) is countable if c a rd ( Y ) < ui. Il J is an arbitrary function, then d o m ( f ) denotes the dom ain of / . Let { X, : i G / } be a fam ily o f nonem pty pairwise disjoint sets and let E = U { Xj : i G / } . We say tha t a set .V Ç E is a partial selector of {,Y; : i 6 / } if, for each index / G I, we have the inequality c n r d ( X П .V,) < 1. Furtherm ore, we say tha t a set А С E is a selecto r o f {A , : г G / } it, for each
i G / , we have the equa lity c u r d ( X П A ,) = 1. O bviously, every partial
selector of {A , : / G /} can be ex tend ed to a selector ot {AT,- : i G / } . Various selectors appear naturally in the situa tion where two groups
Cl and II are given such that II Ç Cl. For instance, let (6\ + ) be a
co m m utativ e group and let II be a subgroup of Cl. W e say tha t a set
X Ç Cl is an //- sele ct o r (respectively, a partial //-selec to r ) if A is a
selector (respectiv ely, a partial selector) of the fam ily Cl/11 canonically associated with I I . In our considerations we suppose, as a rule, that
II is a nontrivial (i.e. nonzero) subgroup of Cl.
Let (Cl,-) be an arbitrary group and let /г be a m easure defined on som e rr-algebra of subsets of Cl. We recall that // is a G'-invariant m easure if d o m ( /i) is invariant with respect to the group of all left translations o f Cl and fi (gY' ) = /<(У ) for each g G Cl and for each У G
d o m ( n ) . A more general concept is the concept of a 6'-quasiinvariant m easure. For a given m easure // on Cl, let us denote by Z( / i ) the class of all /(-m easure zero sots. We recall that // is a (7-quasiinvariant m easure if the classes of sets du m( f i ) and Z( f i ) are invariant with respect to the group of all left transla tions of Cl. ( 'learly, every ('/-invariant m easure is sim ultaneously a (7-q nasi in variant measure. T he converse assertion is not true, in general. Let us observe that if we have a nonzero rr-finite C1- quasiinvariant. (('/-invariant) m easure /< on Cl, then we can easily define a probability (7-quasiinvariant m easure v on Cl such that do rn (v) =
do m( f i ) and Z( i ' ) = Z( f i ) (in other words, ц and v are equivalent
m easures). T his sim ple observation will be applied below.
Of course, one can introduce the concept of a ('/-invariant m easure ((7-quasiinvariant m easure) with respect to the group of all right trans-lations of Cl. If the original group Cl is co m m utative, then two concepts
o f a (7-invariant (G'-quasiinvariant) m easure are identical.
N ow , let us fix an uncountable com m u tative group (6' , + ) with a nonzero (T-finite G’-quasiinvariant m easure fi defined on som e «т-algebra of subsets o f G. In connection with the ab ov em entio n ed V itali theo -rem, the following four questions arise in a natural way.
Q u e s t i o n 1. Let II be an infinite countable subgroup of G . Is it true tha t all //-selecto rs are non m easurable with respect to /<?
Q u e s t io n 2. Let II be a nontrivial countable subgroup o f G. Is it true tha t there ex ists at least one //-sele cto r nonm easurable with respect to //.?
Q u e s t io n 3 . Let II be a countable subgroup of G and let G / H =
{ X i : / ' € / } . Is it true tha t there ex ists a subset J of / such that
all selectors o f the partial fam ily {A', : г G ./} are nonm easurable with respect to //.?
Q u e s t io n 4 . Let II be an uncountable subgroup of G such that
c a r d ( G / / / ) = c a rd ( G ). Is it true tha t there ex ists at least one / / -
selecto r nonm easurable with respect to /<?
One can show that the answer to Q uestion 1 is negative. M oreover, in [6] a. m easure /i is constructed sa tisfy ing the following conditions:
1) fi is defined on som e c-alg eb ra o f subsets o f the real line Ft;
2) //. is a nonzero nonatom ic cr-linite measure;
3) d u w ( A) is contained in d om (fi) ;
t) for each L ebesgue m easurable subset A o f Ft with A(A ) = 0, we have //.(.V) = II;
5) for each Lebesgue m easurable subset A o f ft with \ ( X ) > 0, we have //.( A ) = + o c;
6) // is invariant with respect to the group o f all isom etric transfor-m ations of Fi (in particular, //, is invariant with respect to the group of all translations of Ft);
7) there ex ists a //-m easurable (^-selector.
T hus, we see that there ex ists a V itali subset o f Ft m easurable with respect to a certain nonzero <r-fini te Й-in variant m easure on Ft.
A lso it is not difficult to show th at the answer to Q uestion 4 is n eg a tive (see, e.g ., [7]). Indeed, let us put G = Я 2, where Ft2 is the
Euclidean plane, and let us take as II the subgroup {0 } x R. of (!. E vidently, we have th e equa lities
c a r d ( 0 / l l ) = c( ird(R ) = card((i') = 2W.
D enote by A2 the usual two-dim ensional L ebesgue m easure on (!. Fur-ther, let P be the ir-ideal of subsets of Cl, generated by the fam ily of all /-/-selectors. It can easily be checked that P possesses the following properties:
a) P is invariant under the group of all transla tions of G;
b) A2(.Y) = 0, for each set A" belonging to P , where Af denotes the inner m easure associa ted with A2.
Starting with th ese tw o properties of P and applying the standard argum ent (sre. e.g., [7]), it is not difficult to prove the ex isten ce of a m easure // on G such that
(1) /t is an ex tensio n of A2; (2) // is a G -invariaut m easure; (3) P is contained in dom( f i ) ; (4) f i ( X ) — 0 for each set X (E P.
In particular, we see th at all //-sele cto rs are m easurable with re-spect to // .
R e m a r k I . Let n > 0 be a natural num ber, R n be the //-dim ensional
Euclidean space and let A" denote the //-dim ensional Lebesgue m easure on R n . Finally, let; II be an arbitrary uncountable subgroup of the add itiv e group of R " . Suppose that M artin’s Axiom and the negation of the Continuum H ypothesis are fulfilled. Then it can be proved (see [1]) that there exists a. m easure // on R a sa tisfying the following conditions:
(1) // is an ex tensio n of A"; (2) \i is an /^“-invariant measure;
(3) all //-se le cto r s are m easurable with respect to //; (4) if A" is au arbitrary //-se lec to r, then / t ( X ) — 0.
Now, we are going to show tha t the answers to Q uestions 2 and 3 are positiv e. Moreover, we shall establish a much stronger result (see T h e-orem 1 belo w ). First we shall form ulate several auxiliary propositions.
L e m m a 1. Lei {A’,- : i £ / } b e a fa m i ly o f p a irw ise d isjoint sets
su ch th a t m r d ( X j ) > 1, for all in dices i Ç I, a n d let X b e a p artial selector o f {A'; : i £ I ) . T h e n th ere ex ist tw o selectors Y\ a n d V* o f
{ A’, : i £ /} sa tisfy in g th e e q u a lity Y\ П Yi = X .
T his lem m a is trivia l. From it w e im m ediately obtain th e next proposition.
L e m m a 2 . Let E b e a set e q u ip p e d w ith a m e a su re p a n d let { X j : i £ / } b e a part il ion o f E such th a t c n r d ( X i ) > 1. for all
i £ I . S u p p o s e also that th ere e x ists a p a rtial se lecto r o f {.V, : i £ / } n o n m e a su ra b le w ith resp ect to th e m e asu re /i. T h e n th ere e xists a se le cto r ol { A ; : i £ 1} n o n m e a su ra b le w ith respect to p.
Let (i'i be a group equipped with a probability GVqnasiinvariant m easure / / , . let G2 be another group and let / be an arbitrary hom o-m orphiso-m froo-m (!\ onto (>2• W e denote
S = { Y C G 2 : f - \ Y ) £ d o m ( , h )}.
O bviously, S is a er-algebra o f subsets o f the group (V2, invariant with respect to th e group of all left translations o f (>2■ W e define a functional
p-2 on S by the formula
/<a(V') = Р Л Г ' ( У ) ) ( Y € 4
It is easy to see that the follow ing proposition holds.
L e m m a 3 . //2 is a p ro b a b ility (12 -(piasiin varia n t m e a su re on (72. M o reo ve r. i f th e orig inal m e a su r e //.t is G \ -in v a r ia n t, th en p2 is G 2- in variant.
N otice, in connection with Lem ma that if /i\ is an arbitrary (т-finite G i -«|ita.sii n variaut (respectively, G \ -invariant) m easure ou the group G \ , then the m easure p2 on the group G2, defined liy the sam e form ula, is 6'2-quasiinvariant (respectively, ^ -in v a r ia n t) but we cannot assert, in general, tha t /i2 is <r-fini te.
T he next lem m a plays the key role in our further considerations. L e m m a 4 . Let G be an u n c o u n ta b le c o m m u ta t iv e .g ro u p e q u ip p e d
w it h a n o n zero cr-finite G - q ua siin va rian t m e asu re /«. T h en th ere e xists a su b g ro u p Г o f G n o n m e a su ra b le w ith resp ect to //.
Lem m a 4 was proved in [9]. Here we want to remark only that the proof o f this lem m a is essentially based on som e com binatorial properties o f the Ulam (u.1 x a,’i )-m atrix and on a well known theorem from group theory, concerning the algebraic structure o f com m uta tive groups (m ore precisely, the above-m entioned theorem sta tes th at every co m m utativ e group can be represented as the union of a countable fa m ily o f subgroups each of which is the direct sum o f cyclic groups).
In add ition, we m ay a ssum e in Lem m a 4, w ithout loss of generality, th at th e subgroup Г o f (1 is uncountable. Indeed, it is sufficient to apply Lem m a 4 to any G'-quasiinvariant extension и of g such that
d o m ( v ) contains all counta ble subsets o f G.
From L em m a 4 we can deduce th e following
L e m m a 5 . Let G h e a c o m m u ta t iv e g r o u p w it h a n o nzero о -finite
G -quasi in va ria n t m e a su r e // a n d let II b e a su b g ro u p o f G sa tisfyin g th e i n e q u a lit y c u r d ( G f H ) > и>. T h e n th ere e x is ts a su b g ro u p Г o f G such that
1) I I is c o n ta in e d in Г;
2) Г is n o n m ea su r a b le w ith respec t t o /<.
Proof. W e m ay assum e, w itho ut loss o f generality, th at //. is a prob*
a bility m easure on G . Let us denote by /' the canonical hom om orphism from the given group G on to the factor group G / I I and let us put
•s' = { Y Q G / I I ■■ r ' ( Y ) e d o ,,,( ,,) }.
Further, let us define я m easure и on the rr-algebra .s' by the form ula " ( П = / « ( / - ' ( Y ) ) { V € $ ) .
According to Lem m a 3, /' is a probability (C y //)-q ua siinv a r ia ut m ea-sure on th e uncountable group G / I I . According to Lem m a 4, there ex ists a subgroup Г* o f G / II nonm easurable with respect to v. Let us put Г = ./ —1 ( 1 T hen one can easily verify that Г is a subgroup o f G nonm easurable with respect to // and
II = l t e r ( f ) = _/— 1 (0) С /" '( Г * ) = Г. T hus, L em m a 5 is proved.
L e m m a 6. Let G he an u n co u n ta b le g ro u p e q u ip p e d w ith a n o n zero crUnite G q u asiin va ria n t m e a su r e ft. T h e n th ere e x ists a s u b -set, o f G n o n m ea s u ra b le w ith resp ect to ft.
T his lem m a was proved in [5]. N o tice that a stronger result can be esta blish ed lor rr-hnite invariant m easures (see [11]). B ut the m ethod used in [11] does not work for <r-finite quasiinvariant m easures.
Now, we can form ulate the following result.
T h e o r e m 1. Let G b e an u n co u n ta b l e c o m m u t a t i v e g ro u p e q u ip p e d
w ith a n on zero cr-Iinite G-quasiinvariant, m e a su re // a n d let H be a c o u n ta b le su b g ro u p o f G . D e n o te b y G / I J — {.V; : i G / } th e p a r ti tio n o f G ca no nica lly a sso ciated w ith I I . T h e n th ere e xists a su b se t .1 o f I such th a t
1) th e union o f th e p a rtia l fa m ily {A', : i G J } is a subgroup o f G n o n m ea su ra b le w ith respe ct to ft;
2) all selectors o f {.V, : i € ./} are n o n m e a su ra b le w ith respect to i<;
:i) i f II is a n o n triv ia l su b g ro u p o f G. then there e xis ts an H - sele c to r n o n m e a su ra b le w ith re spect to //..
Proof. A pplying L em m a 5, we see that there ex ists a subgroup
Г
o f G such that
H С Г, Г ^ dom{f t ) . Since Г / I I C ' G/ II, we can write
Г / II = {.Y,-: * € . / } Ç { X r . i e l } ,
for som e .1 Ç / . O bviously, we have the equality r = U { A '
(-C onsequently, relation 1) holds for { A', : ■ / £ . / } . Further, let X b e an arbitrary selector o f {A ; : / € •/}. We assert that X is nonm easurable with respect to //. Suppose otherw ise, i.e. X G dorn(fi ). T hen we have
where all sets h -f Л’ are //-m easurable. Taking into account the fact that H is a countable subgroup o f Cl, we get Г G d o m (/ i) which yields a contradiction. T hus, Л does not belong to d o m( f i ), and relation 2) holds for {.V; : i G ./} . Finally, applying Lemma 2 to the partition ( 7 / / / o f Cl, we im m ediately obtain that relation 2) im plies relation 3). T he proof o f Theorem 1 is co m plete.
R e m a r k I. Unlort unately, T heorem 1 cannot he generalized to the class ol all uncountable groups with nonzero /г-(inite quasiinvariant m easures. Indeed, Shela.li proved in [10] that there ex ists a group (7 with the follow ing properties:
a) card(Cl) = u-’i ;
I)) Cl does not contain a proper uncountable subgroup.
Let us take such a group (1 and let us lix a counta ble subgroup I I o f (1. Further, deno te by N the /т-algebra o f subsets of the group Cl,
generated by the fam ily of all countable subsets of Cl. One can easily define a probability (7-invariant m easure // on S such that //( Y ) = 0
for each countable subset Y o f Cl. Now, it is clear tha t, for (Cl,/i) and / / , an analogue o f T heorem 1 does not hold.
However, we have the following result (cf. [7]).
T h e o r e m 2. Let. (I b e an a rb itra ry u n co u n ta b l e g rou p e q u ip p e d
w ith a n o n ze ro n - fi n it e Cl-<piasiinvariant m e a su re /i, a n d let { X,- : i G / } be a p a rtitio n o f (1 su ch th at
1 < c n r d ( Xj ) < u \
for all in dices / G / . Then tln're e xists at least on e selector o f {.V,- : / G
/ } n o n m ea su ra b le w ith respect to th e m e a su re //. In pa rticular, i f I I
is a n o n trivia l co un t a ble su b g ro u p o f (1 a n d {Л,- : / G / } is an in je ctiv e fa m ily o f all /<■// ( ri ght ) I I- o rb its in Cl. th en th ere e x ists a selector o f
{ A", : i G / } n o n m e a su ra b le w ith resp ect to /i.
Proof. According to Lemm a 6, there is a subset Y o f Cl non-m easurable with respect to //.. Taking into account the inequalities
c u r d { Л t ) 5: w lor all / G I. we easily deduce that the set V' can be
represented in the form
where each set Yn is a partial selector of {.V,- : i G / } . Since V does not belong to (1ит{ц), there ex ists a natural num ber n such that Yu also does not beloug to dom( f i ) . Finally, applying L em m a 2, we conclude th at there ex ists at least one selecto r of { A', : i 6 / } nonm easurable with respect to /t.
R u n ark S. Let E be a set equipped with a m easure /t and let
{.V, : i G /} be a partition of E such that 1 < c a r d ( Xj ) < и for all indices i G I ■ Actually, the argum ent used in the proof of Theorem 2 shows tha t th e next tw o assertions are equivalent:
1) there ex ists a subset of E nonm easurable with respect to /t;
2) there ex ists a selector of {.V; : i G / } nonm easurable w ith respect to //.
We can prove som e analogues of th e preceding results in a more general situ atio n. Nam ely, let G be an uncountable group, S be a ir-ai geh ra of subsets of G and let P be a ir-ideal of subsets of G such tha t
P С S. Suppose also tha t the following relations are fulfilled:
a) .S' is invariant under the group of all left translations o f 6'; b) P is invariant under the group of all left translations of G\ c) the pair (.S'. P) satisfies the Suslin condition (i.e. the co untable chain co ndition).
Then a result sim ilar to Theorem 2 holds for G, (.S', P) and a non-trivial countable subgroup 11 of G. In addition, if G is a com m utativ e group, then a result sim ilar to Theorem 1 holds for G, (.S', P) and a.
counta ble subgroup H of G. T h e proofs of those results are based on the corresponding analogues o f lem m as presented above.
In particular, we can form ulate the follow ing topological result. T h e o r e m 3 . Let G h e an u n co u n ta b le c o m m u ta t iv e g ro u p a n d let
T b e a to p o lo g y on G su ch tha t
a) (G, T ) is a sec on d categ ory topological space;
b) th e tralgehra o f sets h a v in g th e Baire p r o p e rt y in (G , T ) is in -variant u n d er th e g ro u p o f all tra n sla tio ns o f G;
c) th e (т-ideal o f first categ ory sets in ( G , T ) is invariant u n d e r th e g rou p o f all tra nsla tio ns o f G;
d) th e space (G. T ) sat isfies th e S uslin co nd ition (i.e. th e c o u n ta b le chain co n d ition ).
Furt her, let II bt' a c o u n ta b le su b g r o u p o f G a n d let, G / H = {.V; : i. 6 / } b e t h e p a rtit io n o f G c an onically a sso ciated wit.li I I. T h e n there
e x ists a. subset, d o f I su ch th a t
1) th e union o f th e pa rtia l fa m ily {.V, : i £ ./} is a su b g ro u p o f G without, th e Haire p r o p e rt y in (G ,T );
2) all selecto rs o f {ЛГ,- : * € • / } do not. h a ve th e B aire p r o p e rt y in ( G , T ) ;
H) i f II is a n o n t rivia l su b g ro u p o f G , th en t here e x ists an H -s ele c to r without, th e H a i r e p r o p e rt y in (G. T ) .
In th e sim ilar way we can form ulate a topological result analogous to T heorem 2.
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A h h s c i m l r r l \ l i a r a z i s l i r i l i
K O N S T R U K C J A V I T A L I E G O D L A P R Z E M I E N N Y C H G R U P
Z M I A R A M I P R A W I E N I E Z M I E N N I C Z Y M I
W pracy rozważa sic analogou klasycznej konstrukcji V ita lieg o dla nieprzeliczalnych grup przemiennych z ст-sk oń czo ny m i m iaram i prawie niezm ienniczy mi.
I n s t it u t e o f A p p lie d M a t h e m a t ic s U n ive r s it y o f T b ilis i U n iv e r sit y St r. 2, 3 8 0 0 4 3 T b ilis i 4 3 , G e o r g ia
