On density topologies generated by ideals

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 ATHEM ATICA 7, 1995

Jacek Hejd.uk and Aleksander Kharazishvili

O N D E N S I T Y T O P O L O G I E S G E N E R A T E D B Y ID E A L S

We discuss some p ro p e rties of th e den sity topology, g en erated by a given ideal I , in connectio n w ith th e co u ntab le ch ain co n dition . N am ely, we prove th a t for every fin ite fam ily of in v ariant cr-algebras w ith inv arian t er-ideals, satisfyin g the co u n ta b le ch ain co nd itio n , th e re exists an elem ent of th e den sity topology, w hich is n o t m ea-su ra b le w ith re sp ect to all of these <r-algebras. In p artic u la r, we o b ta in a g eneralizatio n of one resu lt given in [5].

Let K be th e real line equipped w ith the sta n d a rd Euclidean topo l-ogy. D enote by I th e usual Lebesgue m easure on K. Let X be an a rb itra ry Lebesgue m easurable subset of R and let x be an a rb itra ry p oin t of R. Take any h > 0 and consider th e real num ber

d ( X , x, h) — l ( X n [x — h, x + h])/2h. S uppose th a t

limh-+ o d ( X , x , h )

the re exists an d denote this lim it by d ( X , x). T he real num ber d ( X , x) is called a density of th e set X a t th e point x. If the equality d ( X , x ) — 1 holds, then the point x is called a Lebesgue density

p oin t of X . T he classical Lebesgue theorem states th a t alm ost all p oints of th e set X are its density points. We m ay p ut

$ d(X ) = { x € R : d ( j r , z ) = l} and we m ay consider the following class of sets:

Td = { X 6 dom (l) : X C $ d(X )}.

It is well-known th a t th e class Td is a topology 011 the set, R , extending th e Euclidean topology of R. T he topology Td is called the density topology of R. It is w orth rem arking here th a t the density topology was intensively investigated by m any au thors (see, for exam ple, [1] where some interesting properties of this topology are considered; see also [2] where some additional properties of the density topology are discussed as well).

It is easy to check th a t the zero-point 0 is a density p oin t of a Lebesgue m easurable set X C R if and only if the relation

lim.n-.oo n ■ l ( X fl [ -1 / n , 1 /n ]) = 2

holds. Obviously, this relation is equivalent to th e equality lim n->oo l ( n X fl [ - 1 , 1]) = 2,

where n X denotes th e set { nx : x (E X }. T he last equality m eans th a t th e sequence of characteristic functions

{XnXn[—1,1] : n € N}

converges in m easure to the characteristic function of the unit, seg-m ent [—1, 1], Now, applying the well-know n Riesz theorem from th e classical m easure theory, we can describe th e convergence in m easure in term s of the convergence alm ost everywhere. T his sim ple (b u t im p o rtan t) observation is due to W .W ilczyński who in tro duce d in 1985 th e concept of a density point w ith respect to category (see [3] and [4]).

(1) th e u n it segm ent [ - 1 , 1] has the B aire p rop erty w ith respect to T;

(2) for every n a tu ra l num ber n and for every set X having th e B aire p ro pe rty w ith respect to T, the set n X has also th e B aire prop erty w ith respect to T;

(3) for every tran slatio n g of R and for every set X having th e B aire p rop e rty w ith respect to T , the set g ( X ) has also th e B aire p ro p e rty w ith respect to T .

N ote th a t these conditions are sufficient to introduce th e concept of a density point in the category sense.

Let B ( R , T ) denote th e a a lg e b ra of all sets having the B aire p ro p -e rty w ith r-esp-ect to th-e topology T and l-et A '(R ,T ) d-enot-e th -e a - ideal of all first category sets w ith respect to T. F urtherm ore, let X be an a rb itra ry set from the <r-algebra B ( R , T ) . We say th a t the z ero-p oint 0 is a K ( R , T )-d e nsity point of the set X if th e sequence of c harac teristic functions

{ /„ : n G N} = { x „ x n [-i,i] : « G N}

is convergent to the characteristic function X[-i,i] w ith respect to th e (7-id e a l A”(R ,T ) . T he last sentence m eans th a t 0 is a K ( R , T ) ~ density point of the set

X

N\

Now, let x be an a rb itrary point of R and let X be an a rb itra ry set from th e <r-algebra B ( R , T ) . We say th a t x is a I \ ( R , T )-d e n sity po in t of the set X if 0 is a Ar (R , T )-d en sity point of th e tra n sla te d set

X - x = { y - x : y £ X } .

F utherm ore, for any set X having the B aire p ro pe rty w ith respect to the topology T, let us p ut

$ T(X ) = th e set of all A'(R, T )-d e nsity points of X . Hence, we ob tain the following family of sets:

If t,he fam ily T* form s a topology on the basic set R , the n T* is called th e W ilczyński topology on R , associated w ith the original topology T (note th a t T* is also called the 7 \( R , T )--density topology on R ).

T h us, we have the concept of a density topology on R in th e sense of category. O f course, an analogous definition can be form ulated for an a rb itra ry group G equipped w ith a topology T satisfying the conditions sim ilar to conditions (1) — (3).

E x a m p l e 1. Let us consider a p articular case of the above con-struction. Namely, let us take as T the sta nd a rd E uclidean topology on R. In this case it can be shown th a t T* is a topology on R called th e I -d e n s ity topology (see [3], [4], and [5]). T he / - density topo l-ogy has a num ber of interesting properties and it was investigated by m any au thors. Am ong various works devoted to th is topology we m ention especially th e book [5] where the class of all continuous functions w ith respect to the I -de nsity topology is studied in details.

E x a m p l e 2. Let us consider ano ther pa rticu la r case of th e W il-czyński construction. Namely, let us p u t T = T j, where Td is the density topology on R. Obviously, T satisfies conditions (1) — (3). It is easy to check th a t the equality

* T ( X ) = * d( X )

holds for each subset X of the real line having the B aire pro p erty w ith respect to T. T hus, we see th a t

T* = T = T d,

i.e. the density topology T d can be considered as a p a rtic ula r case of the W ilczyński topology.

T he next exam ple is a generalization of Exam ple 2.

E x a m p l e 3. Let ^ be an a rb itrary m easure on th e real line, satisfying the following relations:

1) fi is a com plete m easure;

F urtherm ore, let X be a //-m easu rab le subset of R an d let x be a po int of R. We say th a t x is a / / - density point of th e set X if th e equality

n ■ //(X n [x - 1/ n , x + 1/n ]) = 2

holds. It is clear th a t if // = /, then this definition gives us th e classical definition of a Lebesgue density point.

For each set X E dom(f i), let us put

^ ( X ) = the set of all //-density points of X . Now we m ay consider the fam ily of sets

Tft = { X E d o m( n ) : X C $ M(X )}.

It can easily be shown th a t the fam ily T,, is a topology on th e basic set R. Moreover, in [6] is established, using th e classical V itali covering theorem , th a t for any set X E there always exist subsets L , Y and Z of R such th a t

L E d om(l ), n ( Y ) = n (Z ) = 0, X = (L U V ) \ Z.

We see also t h a t the topology extends the usual density topology T d and, if n = /, th en T^ coincides w ith Td.

Suppose now th a t our m easure fi satisfies relations 1), 2) a nd the following two relations:

3) /i is invariant under the group of all tran sla tion s of R;

4) for every n a tu ra l num ber n and for every /¿-m easurable set X , th e set n X is //-m easurable, too, and

n ( n X ) = n • //(X ).

T he n it is easy to see th a t th e notion of a //-d en sity po int can be form ulated in term s of the cr-ideal of all //-m easu re zero sets (by the general schem e of W ilczyński considered above). T hus, we conclude t h a t if relations l ) - 4 ) hold for the given m easure //, th en the topology

can be obtaine d by th e schem e of W ilczyński.

N ote th a t th e W ilczyński construction can be applied in a m ore general case (see, e.g., [5]). Namely, let I be a fixed ideal of subsets of the real line R. Let { / „ : n E N} be a sequence of functions

acting from R into R. We say th a t this sequence converges ( / ) tb a function / : R —> R if for every infinite subset N[ of N the re exists an infinite subset N2 of N i such th a t the pa rtia l sequence of functions if n ■ n e N 2} converges pointw ise to / on the com plem ent of a m em ber from the ideal I.

We say th a t a point x 6 R is an /-d e n s ity point of a given set X C R if th e sequence of characteristic functions

{Xn(X—r )n[—1,1] : n G N} converges ( / ) to the characteristic function

X[-i,i]-D enote by the sym bol <£/(X) the set of all /-d e n s ity p oints of the set X .

Now let us pu t

T/ = { X C R : ! C $ ; ( i ) } .

It is not difficult to check th a t the fam ily T j is a topology on th e set R . We say th a t T/ is the topology on R generated by the given ideal / .

Some general properties of the topology T / are discussed in [5]. In connection w ith these properties a certain set A C R is co nstru cted in [5], satisfying the following relations:

1) A is a Lebesgue non-m e asurab le subset of R;

2) A does not have the Baire pro perty w ith respect to th e E u -clidean topology of R;

3) for each point a 6 A the equality

*ooXn(A — a)n[—1,1] = X[ —1,1]

holds; in p articu lar, A G T j for every ideal / of subsets of R. T h e construction of the set A m entioned above explores essentially in [5] the existence of a Hamel basis of R being also a B ernstein subset of R.

In this pa p er we shall show th a t a m uch stronger result can be obtained. For this purpose we need some auxiliary notions.

Let S be a er-algebra of subsets of the real line R , let J be a cr-ideal of subsets of R and let J c S. Recall th a t the p air (J,-S')

satisfies th e countable chain condition if, for any un countable fam ily {X^ : £ < U i} of pairw ise disjoint sets from 5 , the re exists a set belonging to the <r-ideal J (from this definition it follows also th a t all sets X ( , except a countable num ber of them , belong to th e cr-ideal J ) .

We say th a t a <r-algebra S (respectively, a <7-id e a l J ) is invariant u nd er th e group of all translations of R if for every set X from S (re-spectively, from J ) and for every tra nslatio n g of R , th e set g ( X ) belongs to S (respectively, to J ).

Now, let us consider th e real line R as a vector space E over th e field <Q> of all ra tiona l num bers. According to a w ell-know n theorem of th e theory of vector spaces, there exists a basis B of E (this basis is called a Hamel basis of E ). For any elem ent e € E we have the unique rep re se ntation

e = qi bi + q2b2 + ••• + Qrn bm i

where m — m (e ) is a n a tu ra l num ber, q \, q2, ..., qm are ra tion a l num bers a nd bu b2, •••, bm are pairwise distinct elem ents of B .

Let us p u t

IMI = | i l | + 1^21 + ... + \qm

\-Obviously, the functional || || is a norm on E w ith th e values contained in Q. Moreover, it is easy to see th a t (E , || ||) is a nonseparable norm ed vector space.

Let us take an a rb itra ry sequence

r i , r 2, ... , r k , ... (k e N, k > 0) of strictly positive irration al num bers such th a t

Zzm/t—oo r k = oo.

C onsider the fam ily of sets { A k : k £ N, k > 0}, where A k = {e e E : ||e|| < r k }.

Obviously, each set A k is an open ball in the space E an d, since r k is an irra tio na l num ber, the set E \ A k is open in E , too. These prop erties of th e set A k im m ediately give us the following

L e m m a 1. For the set A k and for each point a o f A k we have the eq uality

H m n—*oo Xn(Ak — a)n[ — 1,1] X[—1,1]*

A n analogous equality holds for the set E \ A k and for each p oint a o f E \ A k .

In particular, for every ideal I o f subsets o f E = M. we have A k € 7 7 , E \ A k E T¡.

A m ore detailed proof of this Lem m a see in [5]. We need also the following

L e m m a 2. For each set A k there exists an uncountable fa m ily {e^ : £ < cji} o f elem ents o f E (certainly, depending on A k) such tha t th e fa m ily

{ A k + : £ < u>i) consists o f pairw ise disjoint sets.

T h e proof of this Lem m a see in [7] where a m ore general result is established. Namely, in [7] is proved th a t if V is an a rb itra ry nonsepa ra ble norm ed vector space and Z is a countable union of balls in V whose radii are equal to a fixed num ber r > 0, th e n there exists a n uncountable fam ily : £ < u?i} of elem ents of V such th a t the fam ily

{ Z + v( : £ < u>i}

consists of pairw ise disjoint sets. From this fact it follows also th a t th e set Z is absolutely negligible in the space V (ab ou t the last notion see [6] or [7]).

L e m m a 3. L et { Ji, J¿, ■■■, Jp j be a finite fam ily o f a -ide a ls o f subse ts o f R , let {S i, 52,.--, 5 P} be a finite fa m ily o f cr-algebras o f subsets o f R , and suppose th at the follow ing relations hold:

V J i

C

C

C

2) all pairs { J i , S i), ( J 2, S2), •••, ( J P, S P) sa tisfy the countable chain condition;

3) all classes o f sets

a re invariant under translations o f

R.

T hen there exists a set A k such that A k $ S i

U

U ... U

Proof. Suppose th a t, for any n a tu ra l num ber k > 0, we have A k

G

U J2 U ... U

D enote by m ( k ) a n a tu ra l num ber from [l,p] such th a t Ak € J m(k)- In this way we o bta in a sequence

m ( l ) , m ( 2 ) , ... , m ( k ), ...

of n a tu ra l num bers belonging to the segm ent [1, p]. Hence, there exists an infinite strictly increasing sequence

ki, k2, fc3, ... of n a tu ra l num bers such th a t

m ( k i ) = m ( k 2) — m { k 3) = ... = in £ [l,p]. Therefore, the relations

•Afcj £ Jmi Akz €E Jmt A k3 £ Jmi •••

are fulfilled. Since we have

lin ik ->oo r k — oo, an d J m is a cr-ideal of sets, we get

R =

U

U

U ... €

which is im possible. Hence, we can conclude th a t there exists at least one n a tu ra l num ber k > 0 such th a t

Now, it is easy to show th a t for this num ber k we also have A k £ S i U S2 U ... U S p.

Indeed, suppose th a t Ak G S m, where m € [l,p], and consider an u nco un tab le fam ily {e¿ : £ < ui\) of elem ents of E = R described in Lem m a 2. Since Ak G S m \ J m and th e classes J m a nd S m are invari-a n t u nd er trinvari-a nslinvari-ation s of R , we deduce th invari-a t invari-all sets of th e disjoint fam ily

{Ak + e$ : £ < u>i}

belong to S m \ J m, too. So, we see th a t the pa ir ( J m , S m ) does not satisfy th e countable chain condition, which c ontradicts relation 2). T hu s, th e proof of Lem m a 3 is com plete.

T aking in to account Lem m as 1, 2 and 3 we can form ulate the following

P r o p o s i t i o n . L et I be an arbitrary ideal o f subsets o f the real line R. L et { J \ , J 2, ... , Jp } be a finite fa m ily o f a -id eals o f subsets o f R and let { S i, S 2, , S p} be a finite fam ily o f a-algebras o f subsets o f R. Supp ose also th at relations 1), 2) and 3) o f L em m a 3 are fulfilled for

5 J2 1 • • • > Zp 1 S1 , S2 5 • • • , Sp.

T hen there exists a subset A o f R such that (1) A G T /, R \ A G T i;

(2) A £ S i U S2 U ... U S p.

T h e pro of of this P roposition can be deduced from the preceding lem m as w ithout any difficulties. Indeed, we m ay p ut A = Ak for a su itable n a tu ra l num ber k > 0.

E x a m p l e 4. C onsider a pa rticu la r case of th e situation described above. Nam ely, let p = 2 and let

J\ — th e cr-ideal of all Lebesgue m easure zero subsets of R; Si = th e cr-algebra of all Lebesgue m easurable subsets of R; J i — th e (7-id e al of all first category subsets of R;

52 = the er- algebra of all subsets of R having th e B aire p rop e rty in R.

Obviously, we have

Ji C

C

th e pairs ( J i , S i ) and ( h i S i ) satisfy the countable chain condition a nd the classes of sets

J

ił

1, 52

are invariant under the group of all tran slatio ns of the real line. F u r-therm ore, let I be an a rb itra ry ideal of subsets of th e real line. T hen, by our proposition, th ere exists a subset A of R such th a t

(1) A E T j , R \ A 6 Ti] (2) A ( £ S X U S 2.

T hus, we ob ta in the result from [5] m entioned above.

We see also th a t the topology Tj does not satisfy the countable chain condition.

Re f e r e n c e s

[1] J .C . O x to by , M easure and category, S p ring er V erlag, B erlin, 1971.

[2] F. T all, T he d en sity topology, Pacific Jo u rn al of M ath em atics 6 2 (1976), 275 - 284.

[3] W . W ilczyński, A generalization o f the d en sity topology, R eal A n aly sis Ex-ch an ge 8 no. (1) (1982—1983), 16-20.

[4] W. W ilczyński, A category analogue o f the density topology, app ro xim ate con-tin u ity , and the approxim ate derivative, Real A nalysis E x ch ang e 1 0 (1984— 1985), 241-265.

[5] K. C iesielski, L. L arson an d K. O staszew ski, I -d e n s ity con tinu ou s fu n c tio n s , M em oirs of th e A m erican M a th em atical Society 1 07 no. 515 (1994).

[6] A .B . K harazishvili, In va ria n t exten sion s o f the Lebesgue m easure, in R u ssian , Izd. T b il. G os. U niv., T b ilisi, 1983.

[7] A .B . K harazishvili, Topological aspects o f m easure theory, in R u ssian , Izd. N aukov a D um k a, Kiev 1984.

[8] J . H ejdu k, A .B . K harazishvili, O n den sity po in ts w ith respect to vo n N e u -m a n n ’s topology, to a p p ea r, R eal A nalysis E xchange.

J . H E JD U K A ND A. K H A RA ZISH V IL I

Jacek Hejduk i Aleksander Kharazishvili

O A B S T R A K C Y J N Y C H T O P O L O G I A C H G Ę S T O Ś C I

W pracy rozw aża się pew ne własności abstrakcyjnych topologii gęstości przy założeniu w arunku przeliczalnego łańcucha. U dowod-niono, że dla dowolnej skończonej rodziny niezm ienniczych a -c ia ł i cr-ideałów spełniających w arunek przeliczalnego łańcucha istnieje ele-m ent abstrakcyjnej topologii gęstości, który nie je st ele-m ierzalny wzglę-dem każdego cr-ciała tej rodziny. W szczególności uzyskano uogólnie-nie re z u lta tu pracy [5].

In stitu te of M ath em atics Łódź U n iversity ul. B an acha 22, 90 - 238 Łódź, P o lan d

In s titu te of A pplied M ath e m a tics U niv ersity of T b ilisi U niversity S tr. 2, 380043 T bilisi 43, G eorgia