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
Marek Dalcerzak
F U N C T I O N S W I T H F I B R E S L A R G E O N E A C H N O N V O I D O P E N S E T
Let X be an in finite set. For ideals I , J C P ( X ) an d a fam ily F C P ( X ) , we give co nd itio ns gu aran teeing th e existen ce of an f : X —* X w hich is co n sta n t on X \ C for som e C € J an d fulfils th e co nd itio n: (*) / - 1 [{*}] C\ V £ I for any * 6 A' an d V € F . T h e re su lt an d its p ro o f are related to th e inv estigatio ns m ad e by H .I. M iller an d W . P ored a. In th e case w hen X fo rm s a pe rfect P olish space an d F con sists of all non void open sets, we stu d y ideals I a d m ittin g an / : X —* X w hich satisfies (*) an d is B orel m easu rab le .
1 . In t r o d u c t i o n
C a ratheo do ry showed in 5 th a t there exists a Lebesgue m easurab le function / : R —> R such th a t f ~ l [E] fl U has positive m easure for each set E of positive m easure and each nondegenerate interval U. A m odified version em ploying the B aire category was ob ta ine d by H. M iller in [7]. He proved the existence of a Lebesgue m easurable / : R —► R such th a t f ~ x [£] D U is of the second category for each set E of second category and each nondegenerate interval U . He even ob taine d (in ZFC ) a stronger result where E in / -1 [2£] fl U is replaced by {a:} (for any x £ R ). T he sam e was shown in [11] in a different way (C ontinuum H ypothesis used th ere can be rem oved
which was observed by K.P.S. B h ashara R ao in [3]). In Section 2 we prove a m ore general result w ith the help of a m ixed m etho d joining the tricks from [7] and [11]. In p artic ular, we get a sim ple proo f in ZF C , good for th e m easure and category cases. Since the re is no un cou ntable disjoint family of m easurable sets of positive m easure (this is th e so-called countable chain condition, abbr. ccc), th ere is no Lebesgue m easurable / : R —► R such th a t 1 [{^}] has positive m easure for each X € R. T he analogous observation can be done for the category case. However, there are n a tu ra l exam ples of ideals J (w hich do not satisfy ccc) ad m itting a Borel m easurable / : R —*■ R whose all fibres are large (i.e. not in I). T h a t property, called (M ), was intro duce d in [2] for ideals of subsets of a perfect Polish space. In Section 2 of th e present paper, we stud y a stronger property, called (M * ), which requires the fibres of / to be large on each nonvoid open set.
In general, we consider ideals I of subsets of an infinite set X a nd always assum e th a t X £ I. A subfam ily H of I is called a base of I if each A £ I is contained in some B (E H. We say th a t two ideals I a nd J are orthogonal if there are B € I an d C G J such th a t B U C = X .
2 . Re m a r k s o n Mi l l e r ’s r e s u l t
R ecall th e following theorem due to Abian and Miller (see [1] an d [7]) which generalizes the result of [12].
T h e o r e m 2 .1 . L et X be a set o f infinite cardinality n. L et A be a fa m ily o f at m o st k subsets o f X , each having cardinality k. D enote by A ( A ) th e fam ily o f all D C X such that U ft D ^ 0 for each U € A . T hen, for each cardinal A < k, the set X can be expressed as th e union o f A pairw ise disjoint sets belonging to A (A ).
T h e o r e m 2.2. A ssum e that I and J are orthogonal ideals o f subsets o f a set X o f cardinality k. Let I have a base H o f size < k and let F C P ( X ) be a given fam ily o f size < k such that | V \ E \ = k for any V E F and E E H . T hen, for each x0 € X , there are a set C £ J and a function f : X —> X such tha t f ( x ) = xq for each x £ X \ C ,
and
(*) / '[i®}] V & I f or a n V x E X and V G F.
Proof. P u t A = { V \ E : V G F and E G H } an d apply T heorem 1.1 to it. T he n X can be expressed as the union of a disjoint fam ily A* C A (A ) of size k. Let X = B U C where B G / , C G J and
B n C — 0. Choose any bijection h : A* —* X and define / : X —► X as follows. If x G B , p u t / ( x ) = Xo, and if x £ B , choose a unique D x G A* such th a t x G D x and pu t f ( x ) = h ( D x ). T hen, obviously, f ( x ) = a;o for x G X \ C. If x G X , then
/ ' -m m ]
(x ) \ B for x / a;0, (x) U B for x = x 0.
C onsider any V G F. Observe th a t V D D ^ I for each D G A (A ). Indeed, if V fl D G I for some D G A (^ ) , we choose E G H such th a t V D D C E. We infer th a t V \ E G A and (V \ E ) fl D = 0, which co ntrad icts the assum ption D G A(.A). Now, taking D = we have h ~ l (x) n V £ I. Since B e l , we get f ~ l [{x}] D V £ I .
In p artic u la r, let X = R and let I (resp. J ) be th e ideal of all Lebesgue null sets (resp. m eager sets) in E. It is well kown th a t the fam ily H of all Gs null sets (resp. F„ m eager sets) form s a base of I (resp. J ) , its cardin ality equals c = |R |, and \V \ E \ = c for any open V ^ 0 and E G H . Moreover, I and J are orthogonal (see [10]). T h us from T heorem 2.2 we derive
C o r o l l a r y 2 .3 . (a) There is an f : R —> R such tha t { x G R : / ( x ) ^
0 } is m eager (thu s f has the Baire pro perty) and / - 1 [ { x } ] f l V has p ositive outer m easure for any x G R and open V ^ 0.
(b ) (see [7], [11]). There is an f : R —> R such that {x G R : / ( x ) ^ 0} is a null set ( thus f is Lebesgue m easurable) and / _1[{x}] D V is o f th e second category for any x G R and open V ^ 0.
A nother inte re sting pa ir of orthogonal ideals to which T heorem 2.2 can be applied is described in [9], Proposition 5.
3. Pr o p e r t y ( M*)
Now, we a dd the requirem ent of the Borel m e asurability of / to condition (*) form ulated in Theorem 2.2. Let X be a perfect Polish space an d I - an ideal of subsets of X . We say (cf. [2]) th a t I has p ro p erty (M ) (resp. property (M *)) if there is a Borel m easurable function / : X —> X such th a t / - 1[{a:}] ^ I for each x G X (resp. / 1 K^}] H F ^ / for any x G X and open V ^ 0). We then say th a t / realizes (M ) (resp. ( M*) ) for I. Obviously, (M *) im plies (M ). We shall show th a t the converse is false (E xam ple 3.5).
Remarks, (a) If I and J are ideals of subsets of X such th a t I C J an d J has (M ) (resp. (M *)), then I has (M ) (resp. (M *)).
(b) Since any tw o perfect Polish space are Borel isom orphic (see [8], 1 G4), we m ay replace f : X —* X in the definition of (M ) an d (M *) by / : X —> Y for a suitable perfect Polish Y .
In [2], several exam ples of ideals with prop erty (M ) are given. O ur aim is to find nontrivial ideals w ith pro pe rty (A/*).
It was noticed in [4], Ex. 1.3, p. 4, th a t there exists a Borel function / from (0 ,1) into (0,1 ) such th a t / _1[{ar}] is dense for each x G (0 ,1) (this was tre a te d as a strong version of the D arboux pro pe rty). T he sam e can be inferred from [2], Th. 3.4, p. 44, where a no the r m ethod leads to a Borel m apping from a perfect Polish space X onto the C an to r space, w ith all fibres dense in X . In fact, the existence of such a m apping im plies th a t the ideal of all nowhere dense sets in X has p rop e rty (M * ). O ur next exam ple of an ideal w ith p ro p erty ( M *) is also derived from [2]. It tu rn s out th a t the respective proof for (M ) given in [2] (generalizing M auldin’s construction from [6]) works for (M *), b u t some p a rts require a m ore detailed analysis which will be done below.
T h e o r e m 3.2 (cf.[2], Th.3.3, p. 42). Let I be a cr-ideal o f subse ts o f a perfect Polish space X . A ssum e tha t I contains all singletons, does n o t contain n on em p ty open sets and has a base consisting o f Gs sets. T hen the a-ideal J o f all sets tha t can be covered by Fa sets from I has p ro p erty (M *).
A non em pty closed set F C X will be called /-p e rfe ct if F fl V ^¿0 im plies F C \V £ I for any open V C X .
Let us explain some notation. Let u> = { 0 ,1 , 2 , .. . } . By 2 <UJ an d 2U' we denote, respectively, the sets of all finite an d infinite sequences of zeros and ones. The em pty sequence (which also be-longs to 2 <u;) will be w ritten as ( ). By s0 and .si we denote the respective extensions of s £ 2 <LJ. For z £ 2W and n E tv, p u t z\n — (z(0), ¿ ( 1 ) , . .. , z(n — 1)). The set 2W, endowed w ith the p ro d -uct topology, is called the C antor space. It form s a perfect Polish space.
T he following lem m a results im m ediately from the co ntru ctio n given in [2], pp. 42-43.
L e m m a 3 .3 . Under the assum ptions o f Theorem 3.2, there is a fam - ily {C " : s £ 2<w, n £ w} o f I-perfect sets with the properties:
(1) for each n o n e m p ty open V C X , there is an n £ cu such tha t
% £ V;
(2) for an y s £ 2<UJ, n £ u and a n o ne m p ty V relatively open in C " , there is an m £ uj such that C lg U C™ C V;
(3) for a ny s £ 2<u' and m £ u j, the condition C ”0l D C™ = 0
holds and there is an n E uj such that U C™ C C " . L e m m a 3 .4 . Under the assum ptions o f T heorem 3.2, i f a fam ily {C " : s E 2 <u>,n £ u>} fulfils conditions ( l)-(2 ) o f L em m a 3.3, then, for an y z £ 2W, a set H £ J and a n on em p ty open V C X , there exists a sequence (n{ : i £ u ) o f nonnegative integers such that
tEu>
Proof. Since H £ J , th ere is a sequence of closed sets Fn £ I such th a t H C T he set V \ Fo is open and no nem p ty (in fact, V \ Fo ^ I since V ^ I and F0 £ I) . By (1), pick n0 £ ui so th a t C f t C V \ F0. For any i £ u j, having n, chosen, pick £ u j so
th a t C C"|’| \ Fi+i (we use (2)); here C ^ \ \ F l+i is none m pty (in fact, it does not belong to I ) and relatively open in C " /. From th e classical C an to r theorem we get C = n « gu> ^ z\ i ^ O f course, C is disjoint from I J ^ j Fn and, consequently, from H .
Proof o f Theorem 3.2. We use the sets C ” from Lem m a 2.3. P u t C a = U ngu) C " for s £ 2<w. T hen we have
(a) Ca0 H C„i = 0 for all s € 2<w, (b) Cs0 U C al C C a for all 5 <E 2<u\
which follows from (3). Define D = rin e w U ig2" C x\n . It is not h ard to prove (see [2]) th a t:
(c) B is a Borel set;
(d) for each x 6 B , there is a unique h( x) € 2“' such th a t x E
P l n g o ; C h ( x ) | n i
(e) the function h : B —* 2W defined in (d) is Borel m easurable; (f ) h- 1 [{;?}] = f | new C z|n for each z E 2“ .
Let g : X —* 2“ be a fixed Borel m easurable extension of h. By (f), we have g ~ l [{z}] 3 D n e w ^ ln - C onsider any nonem p ty open V C X . It suffices to show th a t V D Hngu) ^*|n ^ J • Suppose th a t V nplngu; C z\n = H E J ■ According to Lem ma 3.4, there is a sequence (rii : i E u ) for which 0 ^ ^ z\i \ H - ^ e o ther ha nd
^ n n c i'i - ^ n n c ‘\‘ - h ' tgu;
a contradiction.
By T heorem 3.2, the ideal of sets th a t can be covered by F0
Lebesgue null sets has (M *).
E x a m p l e 3 .5 . Let I consist of all sets A C R such th a t AC \ ( —oo, 0) is of Lebesgue m easure zero and A fl [0, oo) is contained in an F„ set of m easure zero. T h en I form s a a-ideal of subsets of R. O bserve th a t I has pro p erty (M ). Indeed, the family 1+ = { A E I : A C [0,oo)} is a cr-ideal of subsets of X = [0, oo), w hich fulfils th e assum ptions of T heorem 3.2. Hence it has pro perty ( M *) and, consequently, pro p -e rty (M ) (in X ). L-et / + : X —+ R r-ealiz-e pro p-erty ( M ) for I+. If w-e extend / + to a Borel / : R —► R , th en / realizes p rop erty (M ) for I. O n th e o the r ha nd, I has not (M *). Indeed, suppose th a t g : R —► R realizes (A /*) for I. T hen {fif-1[{y}] H ( —oo,0) : y E R} form s an u n -c ountable disjoint fam ily of Borel sets w ith positive m easure, w hi-ch is im possible.
In th e above exam ple, I is not translation-invariant, i.e. the con-dition
A + x E I for any A e I and x G R,
w here A + x — {a + a: : a G A] , is not fulfilled. So, it would be inte resting to find an exam ple om itting th a t fault.
Let us note th a t Exam ple 3.5 essentially uses the fact th a t prop erty ( M ) (unlike (M *)) need not be hereditary w ith respect to open sets. To be m ore precise, let us say th a t an ideal I has p ro pe rty ( M 1) if / fl P ( V ) has ( M ) (in V) for any nonvoid open V C X . Obviously, (M * ) =>■ ( M ') =>■ (M ). O ur exam ple shows, in fact, th a t ( M ) => ( M 1) is false. T his suggests the question w hether ( M f) => ( M m) m u st hold.
A c k n o w l e d g e m e n t . I would like to th an k W . P ored a for her in te re st an d valuable rem arks.
Re f e r e n c e s
[1] A. A bian, P a rtitio n o f nond en um era ble closed sets o f reals, C zech. M a th. J. 26 (1976), 207-210.
[2] M. B alcerzak, S o m e properties o f ideals o f sets in Polish spaces, A c ta Uni- v e rsita tis Lodziensis, thesis, Lódz' 1991.
[3] K .P .S . B h a sk a ra Rao, L etter to W. Poreda, 1991.
[4] A. B ru ckn er, D ifferen tia tio n o f Real Function s, L ecture N otes in M a th em a t-ics, Vol. 659, Springer-V erlag, B erlin, 1978.
[5] C . C a ra th éo d o ry , T heory o f Functions, Vol. 2,, 2nd E nglish ed ., C helsea, New York, 1960.
[6] R .D . M au ld in , P roc. A m er. M ath . Soc. 41 (1973), 535-540, T he B a ire order o f the fu n c tio n s co ntin uou s alm ost everywhere.
[7] H .I. M iller, A n analogue o f a result o f C arathéodory, C as. p ro P est. M at. 106 (1981), 38-41.
[8] Y .N . M oschovakis, D escriptive Set Theory, N orth H olland, A m ste rd a m , 1980. [9] J . M ycielski, S o m e new ideals o f sets on the real line, C olloq. M ath . 2 0 (1969),
71-76.
[10] J.C . O x to b y , M easure and Category, Springer-V erlag, 1971.
[11] W . P oreda , S o m e generalizations o f results o f C arath éodory and Miller, A c ta U niv. L odziensis, Folia M ath em a tica 4 (1991), 117-121.
[12] W . S ierpiński, L ’équivalence pa r decom position fin ite et la m e sure extérieure des ensem bles, F u nd M ath. 3 7 (1950), 209-212.
Marek Balcerzak
F U N K C J E O D U Ż Y C H W Ł Ó K N A C H N A K A Ż D Y M N I E P U S T Y M
Z B I O R Z E O T W A R T Y M
Niech X będzie zbiorem nieskończonym . D la pewnych ideałów J, J C P ( X ) i rodziny F Q P ( X ) uzyskano w arunki dostateczne ist-nienia funkcji / : X —► X stałej n a X \ C dla pew nego C £ J oraz spełniającej warunek: (*) / _1[{-T}] H V $ / dla dowolnych x £ X i V G F. W ynik i jego dowód wiążą, się z wcześniejszymi b a d a -niam i H. M illera i W. Poredy. W przypadku gdy X je st doskonałą przestrzenią polską oraz F składa się z niepustych zbiorow otw artych , bad am y ideały / , dla których istnieje borelowska funkcja / : X —► X spełniająca (*).
In s titu te of M a th em a tics Lódź T echnical U niversity al. P olitechniki 11, 1-2 90-924 Lódź, P o la n d
