Small sets in uncountable Abelian groups

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 ATHEMATICA 7, 1995

Aleksander Kharazishvili

S M A L L S E T S I N U N C O U N T A B L E A B E L I A N G R O U P S

We discuss tw o kinds of sm all sub se ts of an u n co u n ta b le A belian group: negligible sets and absolutely negligible sets. We prove th e existence of som e p a rtitio n s of a given u n cou ntable A b elian gro up w hich con sist of su ch small sets.

In this pa per we consider some finite and countable p a rtitio n s of a given basic set E . These pa rtition s consist of sm all subsets of E . Notice th a t here we discuss only two kinds of sm all subsets of E which are closely connected w ith the general theory of tran sfo rm a tio n groups and quasi-invariant (in pa rticu lar, invariant) m easures w ith respect to such groups. Namely, we investigate here th e so called negligible and absolutely negligible subsets of E . These notions were intro du ced and studied in the works [1] and [2].

Let E be a nonem pty basic space, let T be a fixed group of tra n s -form ations of E and let X be a subset of E . We say th a t the set X is T-negligible in E if the following two relations hold:

1) there exists a probability T-quasi-invariant m easure //. defined on E such th a t X E dom (/i);

2) for every probability T-quasi-invariant m easure A defined on E , if X E dom(A) then X ( X ) = 0.

T he re are m any, essentially different from each other, exam ples of negligible subsets of E (see [1] or [2]). For instance, if E is the Euclidean plane and T is the group of all tra nslation s of E then the gra ph of any real function of a real variable is a T-negligible subset of E .

In connection w ith negligible sets a n a tu ra l question arises: how sm all are these sets? A m ore concrete form of this question is th e following: does there exist a finite p artitio n of the space E into T- negligible sets? It is clear th a t there does not exist a p a rtitio n of E into two T-negligible sets. On the oth er hand, we shall see below th a t some p artitio ns of E into three T-negligible sets are possible.

In fu rth er consideration we restrict ourselves to the case where the basic set E is an uncountable Abelian group identified w ith the group T of all its translatio ns. This case is im p orta n t for various a p -plications. However, let us rem ark th a t in some situ atio ns considered below the condition of com m utativity of the groiip T is not necessary.

We need a num ber of auxiliary facts abou t the stru c tu re of infinite A belian groups.

L e m m a 1. Let ( I \ - f ) be an arbitrary uncountable Abelian group. T hen there exist three subgroups Ti , and T3 o f F such that

1) the cardinality o f each o f these subgroups is equal to l o\ ; 2) th e sum Tj + T2 + T3 is the direct sum o f these subgroups. In connection w ith Lem m a 1 notice th a t it can be proved startin g from th e well known theorem s concerning th e stru c tu re of infinite A belian groups. Indeed, it is known th a t every uncountable A belian group T contains an uncountable subgroup G which is th e direct sum of a fam ily of cyclic groups. B ut it is obvious th a t the group G contains the direct sum of its three subgroups, each of the cardinality

l o\ . Therefore, the same is tru e for the original group F.

Notice th a t this lem m a m ay also be proved independently of the general theory of Abelian groups. Namely, th e required three su b -groups of th e group T can be constructed by th e m ethod of tran sfinite recursion u n til u>i.

In the sequel we need two auxiliary notions. Let T be an A belian group, let G be a fixed subgroup of T and let X be a subset of I \ We say th a t th e set X is finite w ith respect to G if for every elem ent

th e inte rsection (x + G) fi X is a finite subset of I1. Analogously, we say th a t th e set X is countable w ith respect to G if for every elem ent x G T th e intersection (x + G) D X is a countable subset of I \

L e m m a 2. Let T be an arbitrary Abelian group, let G he a fixed uncountable subgroup o f T and let X be a subset o f T finite w ith respect to G. T hen X is a F-negligible subset o f I \

T he proof of Lem m a 2 is not difficult. It can be carried ou t using the m ethods developed in the works [1] and [2]. Let us rem a rk in connection w ith this lem m a th a t if a subset X of an A belian group T is countable w ith respect to some uncountable subgroup G of T the n it is not tru e in general th a t X is T-negligible in I \ T he corresponding counterexam ples can be found in the work [1].

T h e next auxiliary result is due, in fact, to Sierpiński (see, for instance, a well known m onograph [3] of Sierpiński). Notice only th a t in this m onograph Sierpiński form ulates and proves a b e au ti-ful geom etric equivalent of the C ontinuum H ypothesis in term s of a p a rtitio n of th e three-dim ensional Euclidean space in to certain th ree subsets each of which is finite w ith respect to the corresponding axis of coordinates. B ut for our purpose we m ust reform ulate Sierpiiiski’s result m entioned above in term s of a sim ilar p artitio n of the direct sum of any thre e Abelian groups, each of the cardinality u)\. Hence, we do no t need here the C ontinuum Hypothesis.

L e m m a 3. L et T i, Ta and T3 be arbitrary Abelian groups, each o f th e cardinality l ox and let F be the direct sum o f these groups. T hen

there exists a partition o f T in to three sets X , Y and Z such th at 1) the set X is finite w ith respect t o T i;

2) th e set Y is finite w ith respect to F2; 3) the set Z is finite w ith respect to r 3.

In particular, all these three sets are F-negligible in the group F. We need also th e following auxiliary proposition.

L e m m a 4. L et ( I \ + ) be an arbitrary Abelian group and let G be a fixed subgroup o f F . L et Go be a subgroup o f the group G and let X be a su bset o f G finite w ith respect to Go- Suppose also th a t H is a sub set o f the group f which has one-elem ent intersection w ith

every G -orbit in F (in other words, suppose that H is a selector o f the fam ily o f all G -orbits in T). T hen the subset H + X o f the group T is also finite w ith respect to the group Go- In particular, i f the group Go is uncountable then the set H + X is F-negligible in T.

Using th e above lem m as it is not difficult, to prove the following proposition.

P r o p o s i t i o n 1. Let F be an arbitrary uncountable A belian group. T hen there exists a partition o f this group in to som e three F-negligible sets. In particular, for every pro bability F-quasi-invariant m easure H defined on T at least one o f these three sets is non-m easurable w ith respect to fi (in fact, at least tw o o f these three sets are non- m easurable w ith respect to fi).

By th e sam e m ethod a slightly m ore general result th a n P rop osi-tion 1 can be obtained. Namely, if F is an A belian group a nd G is any uncountab le subgroup of T the n there exists a p a rtitio n of th e group r in to th ree G-negligible subsets of T.

Rem,ark 1. Using th e sam e m ethod we can prove a ce rtain topological analogue of P roposition 1. Let F be an a rb itra ry group. Let us consider th e class *S(r) of all topologies T defined on F and satisfying th e following relations:

a) T is a second category space w ith respect to T;

b) th e Suslin num ber c(T) is equal to u , i.e. T satisfies th e countable chain condition;

c) all (left) tra nslation s of F preserve th e ideal of first category sets w ith respect to T and the algebra of sets having the B aire p rop erty w ith respect to T .

Let X be a subset of F. We say th a t this subset is T-negligible in th e topological sense if

1) th ere exists a topology T from the class S^T) such t h a t X has th e B aire p roperty w ith respect to T;

2) for each topology T ' from the class 5(T ), if X has th e B aire p ro pe rty w ith respect to T ' then X is a first category set w ith respect to T '.

Now, a result analogous to P roposition 1 can be form u lated in the following way: for every uncountable A belian group F th ere exists a

p a rtitio n of T into some three T-negligible (in the topological sense) subsets of T. Hence, if T is any topology from the class S ( T ) then at least one of these three subsets does not have the B aire pro p erty w ith respect to T. Moreover, a t least tw o of these three subsets do not have th e B aire prop erty w ith respect to T.

Let us form ulate an oth er proposition ab ou t th e negligible sets which is som etim es useful if we w ant to extend P roposition 1 to th e cases w here the considered uncountable group Y is not necessarily com m utative.

P r o p o s i t i o n 2. L e t I \ an d T2 be any tw o groups and let f be any ho m om orphism from Ti onto T 2. I f a subset Y o f the group T2 is r 2-negligible then the subset X = f ~ l ( Y ) o f the group T x is T i- negligible.

R em ark 2. P rop osition 2 has a direct analogue for th e so called abso-lutely non-m easurable subsets of uncountable groups (in connection w ith this notion see [1], [2] and, especially, [4]). M ore exactly, let T, a n d r 2 be any two groups and let / be any hom om orphism from Ti o n to r 2. If a subset Y of the group T2 is absolutely T2-non- ineasurable in V2 th en the subset X = / ~ 1(50 of th e group Tj is a b -solutely T j-non-m easurable in T j . T here are also some o the r in te re st-ing analogies betw een negligible sets and absolutely non-m easurable sets in unco untable groups.

Let us re tu rn to P roposition 1. It shows, in p artic ula r, t h a t for each unc ountable A belian group T the class of all T-negligible subsets of T is not closed even w ith respect to the finite unions. So, this class is not even a prop er ideal of subsets of T. Therefore, we see th a t T-negligible sets are not, in fact, very sm all subsets of T.

T he following definition describes a certain subclass of the class of all T-negligible sets. It tu rn s out th a t this subclass is a p rop er ideal of subsets of a basic space E .

Let E be a non-em pty basic space, let T be a group of tran sfo r-m a tions of E an d let X be a subset of E . We say th a t th e set X is absolutely T-negligible in E if for every probability T-quasi-invariant m easure fj. defined on E there exists a probability T -quasi-invariant m easure A also defined on E extending ¡.l and such th a t \ ( X ) = 0.

absolutely T-negligible sets is a pro per ideal of subsets of th e space E . So, we m ay conclude th a t no analogue of P ro position 1 can be proved for absolutely T-negligible sets. On the oth er h an d, we shall see below th a t every uncountable A belian group T adm its a coun table p a rtitio n into absolutely T-negligible sets.

We need an auxiliary proposition which yields a purely geom etric characte riza tion of absolutely negligible sets and plays an essential role du ring investigation of these sets.

L e m m a 5. L et E he a non-em pty basic space, let F be a group o f transform ations o f E and let X be a subset o f E . T hen the n e x t tw o relations are equivalent:

1) X is an absolutely T-negligible set in E;

2) for each countable fam ily {</, : i € 1} o f elem ents from the group T there exists a countable fa m ily {h j : j & J ) o f elem ents from T such that

Hj hj(Ui 9i( X ) ) = 0.

For th e proof of Lem m a 5 see [1] or [2]. Let us notice in connection w ith th e result of this lem m a th a t it would be interesting to o b tain a sim ilar purely geom etric characterization of negligible sets.

Using Lem m a 5 it is not difficult to prove the following

L e m m a 6. L et

(r, +)

Now, we can form ulate the following result

P r o p o s i t i o n 3. Let F be an arbitrary uncountable A belian group. T hen there exists a countable partition {X, : i E 1} o f this group in to absolutely F-negligible sets. In particular, for every proba bility T-quasi-invariant m easure ¡i defined on F there exists an in d ex i 6 I (certainly, depending on pi) such that the corresponding set X i is non- m easurable w ith respect to /i. Hence, the m easure /i can be stric tly

ex ten de d to a probability T-quasi-invariant m easure X also defined on F and satisfying the equality \ { X , ) = 0.

Let us m ake some rem arks in connection w ith P roposition 3. In fact, th e result of this proposition is contained in the work [1], while it is no t form ulated there. A short proof of P roposition 3 m ay be done, using th e m ethods of [1], in the following way. A ccording to th e well known general theorem s of the theory of A belian groups (see, for instance, [5]), the uncountable Abelian group T un de r our consideration can be represented as the union of an increasing (w ith respect to inclusion) countable fam ily of subgroups {Tn : n 6 u;} in such a way th a t each subgroup T n is the direct sum of cyclic groups. Now, only tw o cases are possible.

If for every n a tu ra l index n the cardinality of the fam ily of all T orbits in th e group T is strictly greater th an u then, applying L em m a 5, we im m ediately ob tain th a t all sets F n are absolutely T-negligible in th e group T. Using th is fact it is easy to get the required co untable p a rtitio n of T into absolutely T-negligible sets.

Suppose now th a t there exists a n a tu ra l index n such th a t the c ardinality of the family of all Tn-orbits in th e group T is less or equal to u . T he n it is obvious th a t the group r „ is also uncoun table and, m oreover, it can be represented as th e direct sum of two subgroups one of which has the cardinality u>\. Hence, according to th e results of

[1], the re exists a countable p artitio n of the group r „ into absolutely Tn-negligible sets. From this fact, taking into account L em m a 6, we can conclude th a t there exists a countable p a rtitio n of the group T in to absolutely T-negligible sets.

Therefore, in b oth cases we have the required resu lt, and P ro p o -sition 3 is proved.

In connection w ith P roposition 1 and P roposition 3 the following tw o n a tu ra l questions arise.

1. For w hat uncountable groups T an analogue of P ropositio n 1 is true?

2. For w hat uncountable groups P an analogue of P ropositio n 3 is true?

These questions are still open. Of course, the second question is m ore im p o rta n t from the point of view of the theory of quasi-invariant

(in p a rticu la r, invariant) m easures. Notice th a t if th e cardinality of the group T is equal to then we have a direct analogue of P rop ositio n 3 for this group. More generally, if E is a basic space of the card inality ui\ and T is a transitiv e group of transform a tio ns of E acting freely on E then there exists a countable p a rtitio n of E in to absolutely T-negligible sets (see [1] or [2]).

We also w ant to rem ark here th a t all the results above rem ain valid if we consider the class of non-zero cr-fmite quasi-invariant m ea-sures in stead of the class of probability quasi-invariant m eaea-sures. M oreover, these results rem ain valid for the class of quasi-invariant (in p a rticu la r, invariant) m easures which satisfy th e Suslin condition (i.e. th e countable chain condition). Finally, these re sults show also th a t m any facts of the theory of T-quasi-invariant (in p a rtic ula r, F- inva rian t) m easures are connected only w ith the algebraic stru c tu re of th e group T of transform ations of the space E.

At the end of this pa per let us consider a topological application of P roposition 3. Let T be an a rbitra ry group. Denote by th e sym bol O(T) th e class of all topologies T defined on T and satisfying the following thre e relations:

a) T is a B aire space topology on T;

b) the Suslin num ber c ( T ) is equal to u j, i.e. T satisfies th e countable chain condition;

c) all (left) translations of the group T preserve the ideal of first category sets w ith respect to T and the algebra of sets having th e B aire p rop erty w ith respect to T.

In p a rtic ular, if (T, T ) is any er-compact locally com pact topologi-cal group then , of course, the topology T belongs to the class 0 ( r ) .

We have th e following result.

P r o p o s i t i o n 4. Let F be an arbitrary uncountable A belian group and let T be any topology from the class O(T). T hen there exists a topology T ' in this class such that

1) T ' strictly extends T;

2) the ideal o f first category sets w ith respect to T is stric tly contained in the ideal o f first category sets w ith respect to T '. T he proof of P roposition 4 can be obtained using the result of

P rop ositio n 3. Indeed, we have a countable p a rtition {.Y, : i € 1} of the group T into absolutely T-negligible sets. Obviously, at least one set X{ is not a first category subset of T w ith respect to th e original topology T . So, we can extend bo th the topology T and the ideal of first category sets w ith respect to T using the m entioned set X{.

Re f e r e n c e s

[1] A .B . K h arazishvili, S om e questions o f Set theory and M easure the o ry , (in R u ssian ), Izd. T b il. Gos. U niv., T bilisi (1978).

[2] A .B . K harazishvili, In varia nt extensions o f the Lebesgue m easure, (in R us-sian ), Izd. T b il. G os. U niv., T bilisi (1983).

[3] W . Sierpiń ski, C ardinal and O rdinal n um b ers, P W N , W arszaw a, 1958. [4] A .B. K harazishvili, A bsolutely non-m easurable sets in A belian groups, (in

R u ssian ), Soob. A cad. N auk G ruz. SSR 97 no. 3 (1980).

[5] A .G . K uro sh , T he theory o f Groups, vol. 1, C helsea P u blish in g C om p an y , New York, 1955.

Aleksander Kharazishvili

M A Ł E Z B IO R Y W N I E P R Z E L I C Z A L N Y C H G R U P A C H A B E L O W Y C H

W pracy rozważa się pewne własności m ałych i absolutnie m ałych zbiorów w nieprzeliczalnych grupach abelowych.

In stitu te of A p plied M a th e m atics U niv ersity o f T bilisi U niversity S tr. 2, 380043 T bilisi 43, G eo rgia