On Borel maps, calibrated σ-ideals and homogeneity

Transcription

1 On Borel maps, calibrated σ-ideals and homogeneity Institute of Mathematics University of Warsaw Ideals and exceptional sets in Polish spaces, Lausanne, 4-8 June 2018

2 The results come from a joint paper with Roman Pol under the same title, to be published in Transactions of the AMS.

3 Homogeneous σ-ideals Following Zapletal [Forcing idealized], we say that a σ-ideal I on a Polish space X is homogeneous, if for each Y P BorpX qzi there exists a Borel map f : X Ñ Y such that f 1 paq P I, whenever A P I. Example The following σ-ideals are homogeneous: the σ-ideal of countable subsets of (an uncountable) Polish space X, the category σ-ideal of meager Borel sets in the Cantor set, the measure σ-ideal of Lebesgue-null Borel sets in the Cantor set.

4 Two σ-ideals related to a measure By a Borel measure on X we mean a countably additive measure µ : BorpX q Ñ r0, 8s defined on the σ-algebra of Borel sets in X such that µpx q ą 0. µ is σ-finite if X is the union of a countable family of Borel sets with finite µ-measure. µ is semifinite if each Borel set in X of positive µ-measure contains a Borel set of finite positive µ-measure.

5 Two σ-ideals related to a measure Given a Borel nonatomic measure µ on a compact metric space X we shall consider the following σ-ideals: J 0 pµq the σ-ideal of Borel sets in X that can be covered by countably many compact sets of µ-measure zero, J f pµq the σ-ideal of Borel sets in X that can be covered by countably many compact sets of finite µ-measure.

6 Examples of Borel measures on compact metric spaces Example Let λ be the Lebesgue measure on r0, 1s or on the Cantor space t0, 1u N. Then J 0 pµq E. Let H 1 be the 1-dimensional Hausdorff measure on the square r0, 1s 2, i.e., for ε ą 0 and E P Borpr0, 1s 2 q let H 1 ε peq inf! ř n diampu nq : diampu n q ď ε, E Ď Ť n U n H 1 peq lim εñ0 H 1 ε peq. H 1 is a Borel, semifinite, non-σ-finite nonatomic measure on r0, 1s 2 and there exists a dense G δ subset of r0, 1s 2 of H 1 -measure zero. ),

7 Non-homogeneity of J 0 pµq Proposition Let µ be a semifinite nonatomic Borel measure on a compact metric space X. Then for each Y P BorpX qzj 0 pµq with µpy q 0 and any Borel map f : X Ñ Y there is a compact set C in Y with µpcq 0 but f 1 pcq R J 0 pµq. In particular, the σ-ideal J 0 pµq is not homogeneous. Proof. By the Lusin theorem, there is a compact set K in X with µpkq ą 0 such that f K is continuous. If C f pkq, then C P J 0 pµq but f 1 pcq R J 0 pµq.

8 Non-homogeneity of J f pµq Proposition Let µ be a semifinite nonatomic Borel measure on a compact metric space X. If µ is not σ-finite and there exists a Borel set Y R J f pµq with µpy q ă 8 and µ KpX q is a Borel mapping on the hyperspace KpX q, then the σ-ideal J f pµq is not homogeneous. Proof. Pick Y P BorpX qzj f pµq with µpy q ă 8. For any Borel function f : X Ñ Y there is a compact set K in X with K R J f pµq (even: of non-σ-finite measure) such that f K is continuous. A reason: the σ-ideal generated by all Borel sets of finite µ-measure is polar, i.e., it is the intersection of the measure σ-ideals of a family of finite Borel measures on X.

9 J 0 pλq and J f ph 1 q are not homogeneous Corollary 1 J 0 pλq is not homogeneous, 2 J f ph 1 q is not homogeneous.

10 A homogeneity result for J 0 pλq J 0 pλq is not homogeneous: for any Borel map f : B Ñ Y from any Borel B with λpbq ą 0 into any Borel null-set Y R J 0 pλq there is a compact C Ď Y with λpcq 0 but f 1 pcq R J 0 pλq. Theorem There is a copy of the irrationals P in r0, 1s such that P R J 0 pλq, for each Y P Borpr0, 1sqzJ 0 pλq there is a homeomorphic embedding h : P Ñ Y such that, for A P BorpPq, A P J 0 pλq if and only if hpaq P J 0 pλq. Corollary The completion of the quotient Boolean algebra Borpr0, 1sq{J 0 pλq is homogeneous.

11 A homogeneity result for J f ph 1 q A reminder: J f ph 1 q is not homogeneous. Theorem There is a copy of the irrationals P in r0, 1s 2 with the following properties: P R J f ph 1 q, for each Y P Borpr0, 1s 2 qzj f ph 1 q there is a homeomorphic embedding h : P Ñ Y such that, for A P BorpPq, A P J f ph 1 q if and only if hpaq P J f ph 1 q. Corollary The completion of the quotient Boolean algebra Borpr0, 1s 2 q{j f ph 1 q is homogeneous.

12 Homogeneity of P I versus homogeneity of I Given a σ-ideal I on a Polish space X denote by P I the partial order of Borel subsets of X not in I ordered by inclusion. Zapletal (see Forcing idealized ): in all cases encountered in his monograph the homogeneity of the forcing associated with P I and the homogeneity of the underlying ideal I always come together. The results above show that for J 0 pλq and J f ph 1 q this is not the case.

13 A σ-ideal related to dimension Remark I pdimq is the σ-ideal of Borel sets in the Hilbert cube r0, 1s N that can be covered by countably many finite-dimensional compact sets, For a compact set X Ď r0, 1s N with X R I pdimq, I X pdimq is the σ-ideal I pdimq restricted to BorpX q. For any compact set X Ď r0, 1s N with X R I pdimq, I X pdimq adds infinitely equal real (J. Zapletal, Dimension theory and forcing, Topology Appl. 167 (2014), 31-35) but no Cohen real (R. Pol, P. Zakrzewski, On Borel mappings and σ-ideals generated by closed sets, Adv. Math. 231 (2012), ). The σ-ideals I X pdimq (as well as forcings associated with them) could be very much different.

14 Calibrated σ-ideals Definitions A σ-ideal I is calibrated if it is generated by compact sets in X and for any K P KpX qzi and K n P I X KpX q, n P N, there is a compact set L Ď Kz Ť K n not in I. npn Examples of calibrated σ-ideals: the σ-ideal of countable subsets of (an uncountable Polish space) X, J 0 pµq for a semifinite Borel measure µ on a Polish space X, I X pdimq for a certain X Ď r0, 1s N (it suffices that X is infinite-dimensional and all of its compact subsets are either infinite-dimensional or zero-dimensional, i.e., X is a Henderson compactum).

15 Calibrated σ-ideals Definitions A σ-ideal I is calibrated if it is generated by compact sets in X and for any K P KpX qzi and K n P I X KpX q, n P N, there is a compact set L Ď Kz Ť K n not in I. npn Examples of non-calibrated σ-ideals: the category σ-ideal, J f ph 1 q (there is a H 1 -null G δ set P Ď r0, 1s such that P R J f ph 1 q but PzP P J f ph 1 q), I pdimq (there is a compact set Y Ď r0, 1s N not in I pdimq containing a zero-dimensional G δ -set G such that Y zg P I pdimq; more precisely, there is a base for the topology of Y whose elements have boundaries in I pdimq).

16 I pdimq is not homogeneous in a strong sense Proposition There are compact sets X, Y in r0, 1s N not in I pdimq such that for any B P BorpX qzi pdimq and every Borel map f : B Ñ Y there is a zero-dimensional compact set C in Y with f 1 pcq R I pdimq. More precisely, the above holds true for X such that I X pdimq is calibrated and Y R I pdimq such that there is a base for the topology of Y whose elements have boundaries in I pdimq. In particular, I pdimq is not homogeneous. Note: non-homogeneity of I pdimq requires only to consider Borel maps f : r0, 1s N Ñ Y (equivalently f : X Ñ Y ), no pattern P (below X ) in contrast with the case of J 0 pµq and J f pµq.

17 A general result Definitions Theorem Let I be a calibrated σ-ideal on a compact metric space X without isolated points, containing all singletons, and let f : B Ñ Y be a Borel map from B P BorpX qzi to a compact metric space Y without isolated points. Then (i) if Y R I pdimq has a basis for the topology whose elements have boundaries in I pdimq, then there is a zero-dimensional compact set C in Y with f 1 pcq R I ; (ii) there exists a compact meager set C Ď Y with f 1 pcq R I ; (iii) for any σ-finite nonatomic Borel measure µ on Y there is a compact set C in Y with µpcq ă 8 and f 1 pcq R I.

18 A general result remarks Remark Recall that (i) states that if Y R I pdimq has a basis for the topology whose elements have boundaries in I pdimq, then there is a zero-dimensional compact set C in Y with f 1 pcq R I. This applied to I I X pdimq for an X Ď r0, 1s N such that I X pdimq is calibrated shows that I pdimq is not homogeneous in a way which implies that the forcings associated with the collections BorpX qzi pdimq and BorpY qzi pdimq, partially ordered by inclusion, are not equivalent. This provides an answer to a question by Zapletal.

19 A general result remarks Remark Recall that (ii) states that there exists a compact meager set C Ď Y with f 1 pcq R I. This shows that for any calibrated σ-ideal I on a compact metric space X without isolated points, containing all singletons, the forcing associated with the collection Bor px qzi does not add Cohen reals, I has the 1-1 or constant property of Sabok and Zapletal: whenever f : B Ñ Y is a Borel map on B P BorpX qzi into a Polish space Y with all fibers in I, then there exists C P BorpBqzI on which f is injective. This strengthens a result from our previous paper which stated the same under assumption that I X KpX q is coanalytic in KpX q.

20 A general result remarks Remark Recall that (iii) states that for any σ-finite nonatomic Borel measure µ on Y, there is a compact set C in Y with µpcq ă 8 and f 1 pcq R I. The σ-ideal J 0 pµq is calibrated, and hence, if µ is σ-finite and nonatomic, (iii) implies that for any Borel map f : B Ñ X on B P BorpX qzj 0 pµq, there is a compact set C in X with µpcq ă 8 and f 1 pcq R J 0 pµq; this implies that if, moreover, X R J f pµq, then the forcings associated with the collections BorpX qzj 0 pµq and BorpX qzj f pµq are not equivalent.

21 Generalized Hurewicz systems A generalized Hurewicz system in a given non-empty G δ -set G in X is a pair pu s q spn ăn, pl s q spn ăn of families of subsets of X with the following properties (the diameters are with respect to a fixed complete metric on G and the closures are taken in X ): U s Ď G is relatively open, non-empty and diampu s q ď 2 lengthpsq, U s X U t H for distinct s, t of the same length, U sˆi X G Ď U s, L s Ď U s is compact, L s X U sˆi H, L s Ş Ť j iąj U sˆi, lim i diampu sˆi q 0.

22 Generalized Hurewicz systems If pu s q spn ăn, pl s q spn ăn is a generalized Hurewicz system, then P č n ď tus : length(s) nu is the G δ -subset of G (actually, a copy of the irrationals) determined by the system and we have, P P Y Ť tl s : s P N ăn u and the sets forming the union are pairwise disjoint, if V is a non-empty relatively open subset of P, then V contains L s with arbitrarily long s P N ăn.

23 A reminder of a homogeneity result for J f ph 1 q Theorem There is a copy of the irrationals P in r0, 1s 2 with the following properties: P R J f ph 1 q, for each Y P Borpr0, 1s 2 qzj f ph 1 q there is a homeomorphic embedding h : P Ñ Y such that, for A Ď P, A P J f ph 1 q if and only if hpaq P J f ph 1 q.

24 How to define P? Definitions Denote µ H 1. Pick a dense G δ copy of the irrationals G in r0, 1s 2 such that µpgq 0. For any non-empty relatively open set U in G we have µpuzgq 8. Fix such a U. By a theorem of Gelbaum, there is a Cantor set L Ď UzG (in particular, L is boundary in U) with µplq 1. Construct nonempty relatively clopen subsets V i of U such that V i X G Ď U, V i are pairwise disjoint and disjoint from L, lim i diampv i q 0 and L Ş Ť V i. n iěn Repeat the above construction at each step of an inductive definition of a generalized Hurewicz system pu s q spn ăn, pl s q spn ăn in G such that for each s P N ăn, L s Ď U s zg is a Cantor set with µpl s q 1.

25 Why does P work? Definitions Recall that if V is a non-empty relatively open subset of P, then V contains infinitely many sets L s with µpl s q 1. It follows, by the Baire category theorem, that P R J f pµq. Fix Y P BorpX qzj f pµq. The idea: construct a copy P 1 of P inside Y. More precisely, P 1 will be determined by a generalized Hurewicz system pu 1 sq spn ăn, pl 1 sq spn ăn with the same properties as above. The construction is carried out in a copy of the irrationals G 1 in Y such that µpg 1 q 0 but U 1 R J f pµq hence µpu 1 zg 1 q 8 for any non-empty relatively open set U 1 in G 1. A G δ subset of Y not in J f pµq can be found with the help of a theorem of Solecki.

26 Why does P work? Definitions The system pusq 1 spn ăn, pl 1 sq spn ăn in G 1 is constructed together with associated homeomorphisms h s : L s Y ď i U s i Ñ L 1 s Y ď i U 1 s i, satisfying the following conditions for each s P N ăn : h s pu s i q Us i 1, h s L s : L s Ñ L 1 s preserves µ. The construction is based on the following

27 Why does P work? Definitions Proposition Let U and U 1 be non-empty relatively open sets in G and G 1, respectively. Let L Ď UzG be a Cantor set with µplq 1. Then there exist a Cantor set L 1 Ď U 1 zg 1 and a homeomorphism g : U Y L Ñ U 1 Y L 1 such that g L is a µ-preserving homeomorphism between L and L 1. The proof of the proposition uses the theorems of: Oxtoby, providing a Cantor set L 1 Ď U 1 zg 1 and a µ-preserving homeomorphism f : L Ñ L 1. Pollard, which, since U Y L and U 1 Y L 1 are copies of irrationals, provides an extension of f to a homeomorphism g : U Y L Ñ U 1 Y L 1.

28 Why does P work? Definitions If f P N N, both Ş m U f m and Ş m U1 f m are singletons, which gives rise to a homeomorphism h : P Ñ P 1, defined by letting hpxq P Ş m U1 f m for x P Ş m U f m. It turns out that for any relatively closed set C in P and s P N ăn, h s pc X L s q hpcq X L 1 s, and in effect, µpc X L s q µphpcq X L 1 sq. It follows that µpcq µphpcqq and this shows that both h and h 1 take sets from J f pµq to sets in J f pµq, completing the proof.

29 A general result Definitions Theorem Let I be a calibrated σ-ideal on a compact metric space X without isolated points, containing all singletons, and let f : B Ñ Y be a Borel map from B P BorpX qzi to a compact metric space Y without isolated points. Then (i) if Y R I pdimq has a basis for the topology whose elements have boundaries in I pdimq, the there is a zero-dimensional compact set C in Y with f 1 pcq R I ; (ii) there exists a compact meager set C Ď Y with f 1 pcq R I ; (iii) for any σ-finite nonatomic Borel measure µ on Y, there is a compact set C in Y with µpcq ă 8 and f 1 pcq R I ;

30 A general result the idea of a proof In each case, assume to the contrary and construct a suitable Hurewicz system pu s q spn ăn, pl s q spn ăn such that if P is the set determined by this system, then: P R I, f ppq is small (zero-dimensional, meager, of finite measure, respectively). To guarantee that P R I just let L s R I for each s P N ăn and use a Baire category argument.

31 How to guarantee that f ppq is small? Using a theorem of Solecki, find a non-empty, G δ -set G Ď B such that V R I for any non-empty relatively open set V in G, f G : G Ñ Y is continuous. Given a relatively open set U H in G consider the maps q fu : Y Ñ KpUq and q f U : KpY q Ñ KpUq defined by q fu pyq Ş n f 1 pbpy, 1 nqq X U, for y P Y, (note: x P q f U pyq if and only if there is a sequence px n q of elements of U such that lim n x n x and lim n f px n q y). q fu res Ť t q f U pyq : y P Eu for E P KpY q.

32 How to guarantee that f ppq is small? Lemma (Main lemma) Let pj s q spn ăn be a family of hereditary collections of closed subsets of Y. Assume that for every non-empty relatively open set U in G and each s, there exist compact sets L and M P J s with the following properties: (1) L is boundary in U and L R I, (2) f 1 pmq P I, (3) L Ď q f U rms, i.e., for each x P L there is a sequence px n q of elements of U such that lim n x n x and lim n f px n q P M.

33 How to guarantee that f ppq is small? Lemma (main lemma continued) Then there exists a generalized Hurewicz system pu s q spn ăn, pl s q spn ăn with an associated family pm s q spn ăn of compact sets such that if P is the set determined by the system, then for each s P N ăn : (4) L s R I hence P R I, (5) M s P J s and M s Ď f pu s q, (6) M s Ş Ť f pu s i q, n iěn (7) f ppq f ppq Y Ť tm s : s P N ăn u.

34 How to construct a suitable Hurewicz system? For every U satisfying (1) (3) there exist nonempty relatively open subsets V i of U, i P N, such that V i X G Ď U, V i are pairwise disjoint and disjoint from L, L Ş Ť V i, n iěn f pv i q are pairwise disjoint and disjoint from M, lim diampv iq 0 and lim diampf pv i qq 0, iñ8 iñ8 Ş Ť f pv i q Ď M. n iěn

35 How to construct a suitable Hurewicz system? Sketch of proof. Fix a countable dense set in L ta 0, a 1,...u where each point is listed infinitely many times. Choose inductively non-empty relatively open sets V i in U such that, V i Ď Bpa i, 1 i`1 q, V i X G Ď U, diampf pv i qq ď 1 i`1, f pv iq Ď BpM, 1 i`1 q, V i Ď UzpL Y Ť jăi V jq, f pv i q Ď f puqzpm Y Ť jăi f pv jqq. Suppose that V j, j ă i, are already defined.

36 How to construct a suitable Hurewicz system? Since a i P L, by (3) we have a i P q f U pb i q for some b i P M. Let δ i ă 1 i`1 be such that Bpa i, δ i q X Ť jăi V j H, Bpb i, δ i q X Ť jăi f pv jq H. Since a i P q f U pb i q it follows that W Bpa i, δ i q X f 1 pbpb i, δ i qq X U H, The function f G being continuous, W is relatively open in U. Since L is boundary in U, W zl H and, moreover, W zl R I as a non-empty relatively open subset of G. Since f 1 pmq P I, we can pick c P W zpl Y f 1 pmqq. Then f pcq P Bpb i, δ i qzm and by continuity of f G, there is a relatively open neighbourhood V i of c in U with V i Ď Bpa i, δ i qzl, V i X G Ď U and f pv i q Ď Bpb i, δ i qzm.

37 How it works for the measure case? Theorem Let I be a calibrated σ-ideal on a compact metric space X without isolated points, containing all singletons, and let f : B Ñ Y be a Borel map from B P BorpX qzi to a compact metric space Y without isolated points. Then for any σ-finite nonatomic Borel measure µ on Y, there is a compact set C in Y with µpcq ă 8 and f 1 pcq R I ;

38 How it works for the measure case? Sketch of proof. Assume to the contrary that f 1 pcq P I for any compact set C in Y with µpcq ă 8. Since the measure µ is σ-finite, there are compact sets F i in Y with µpf i q ă 8 and such that if we let H Y z Ť F i, then µphq 0. i We have f 1 pf i q P I for each i, so Bz Ť f 1 pf i q R I and using a i theorem of Solecki, we find a non-empty G δ -set G in X, G Ď Bz Ť f 1 pf i q, such that V R I for any non-empty relatively i open V in G and f G : G Ñ H is continuous.

39 How it works for the measure case? Lemma Let U be a non-empty relatively open set in G and let ε ą 0. Then there exist compact sets L Ď U and M Ď f puq such that (1) L is boundary in U and L R I, (2) µpmq ă ε, (3) L Ď q f U rms.

40 How it works for the measure case? To prove the lemma, consider two cases. Case 1. There exists i such that q f U rf i s R I. We can cover F i by finitely many compact sets M 0,..., M n 1 with µpm j q ă ε for each j. Then for some j, q f U rm j s R I. We let M M j and pick a compact set L Ď q f U rms, not in I and boundary in U. Case 2. For all i, q f U rf i s P I. I being calibrated, pick a compact set L Ď Uz Ť q fu rf i s not in I i and boundary in U. Let M ty P f puq : q f U pyq X L Hu. Consequently, L Ď q f U rms and it suffices to see that µpmq ă ε. But since L X Ť q fu rf i s H, if y P M, then y P Y z Ť F i H. i i But µphq 0, so µpmq 0.

41 How it works for the measure case? We can now apply the main lemma with J s being the collection of compact sets M in Y with µpmq ă 1, where e : N ăn Ñ N is a 2 epsq fixed bijection. We get P R I such that f ppq f ppq Y Ť tm s : s P N ăn u. Since f ppq Ď H where µphq 0 and µpm s q ă 1 2 epsq for each s P N ăn, we have µpf ppqq ď 2. In effect, for C f ppq we have µpcq ă 8 but f 1 pcq R I which contradicts our assumptions and ends the proof.

42 Thank you for your attention