Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set

Maxim R. Burke ^{∗}

Department of Mathematics and Statistics University of Prince Edward Island Charlottetown PE, Canada C1A 4P3 burke@upei.ca

Arnold W. Miller ^{†}

University of Wisconsin-Madison Department of Mathematics, Van Vleck Hall 480 Lincoln Drive Madison, Wisconsin 53706-1388 miller@math.wisc.edu http://www.math.wisc.edu/∼miller

Abstract

We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A ∩ C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.

1 Introduction

Our starting point is the following question of Laczkovich:

Does there exist (in ZFC) a nonmeager set that is relatively meager in every nowhere dense perfect set?

Note that the continuum hypothesis implies the existence of a Luzin set, i.e., an uncountable set of reals which meets every nowhere dense set in a countable set. Hence, we can think of Laczkovich’s question as asking whether one can construct a particular weak version of a Luzin set without any extra set theoretic assumptions.

Recall that a space X is Polish iff it is completely metrizable and separable. A subset of X is nowhere dense iff its closure has no interior and it is meager iff it is the countable union of nowhere dense sets. A subset of X is residual iff it is the complement of a meager set. A perfect set in

∗Research supported by NSERC. The author thanks the Department of Mathematics at the University of Wisconsin for its hospitality during the academic year 1996/1997 when the earlier result mentioned in the introduction was produced and F.D. Tall and the Department of Mathematics at the University of Toronto for their hospitality during the academic year 2003/2004 when the present paper was completed.

†Thanks to the Fields Institute for Research in Mathematical Sciences for their support during part of the time this paper was written and to Juris Steprans who directed the special program in set theory and analysis.

AMS Subject Classification: Primary 03E35; Secondary 03E17 03E50.

Key words and phrases: Property of Baire, Lebesgue measure, Cantor set, oracle forcing

a Polish space is a closed nonempty set without isolated points, and a Polish space is said to be perfect if it is nonempty and has no isolated points. As we shall see, the underlying space in the question of Laczkovich can be taken to be any perfect Polish space. If we ask, as is quite natural, for the nowhere dense perfect sets in the statement to be Cantor sets (i.e., sets homeomorphic to the Cantor middle third set), then we do not know whether the nature of the Polish space matters. Even for various standard incarnations of the reals (the real line, the Baire space, and so on), we have only partial results on their equivalence in this context. We answered Laczkovich’s question for the Cantor set in 1997 by building a model where the answer is negative. (And of course the perfect nowhere dense sets in this case are necessarily Cantor sets.) Very shortly afterwards, we noticed the more elegant solution presented here which uses a slightly stronger variant of a statement proven consistent by Shelah in [Sh1980]. We show in Section 3 that the stronger conclusion in which, for any perfect Polish space, the perfect nowhere dense sets can be taken to be Cantor sets follows from yet another variant on the same statement. The proof of the consistency of the variants in question is similar to the proof of Shelah. Unfortunately, the proof is quite technical and the argument in [Sh1980] is only a brief sketch, so we give the argument in some detail in Section 4 in order to be clear. An alternative model for the negative answer to Laczkovich’s question for the Cantor set is provided by a paper of Ciesielski and Shelah [CS]. See Remark 3.6. In the final section of the paper, we show how a measure theoretic version of our results can be deduced from results in Roslanowski and Shelah [RS]. The authors thank Ilijas Farah for helpful discussions concerning the models constructed in [RS].

Write perfect(X) for a Polish space X to mean that for every nonmeager set A ⊆ X there is a nowhere dense perfect set P ⊆ X such that A ∩ P is nonmeager relative to P. Write cantor(X) if moreover P can be taken to be a Cantor set. Note that perfect(X) and cantor(X) are trivially equivalent in spaces in which nowhere dense perfect sets are necessarily Cantor sets, e.g., 2^{ω} and ℝ.

We recall for emphasis the following well-known elementary fact of which we will make frequent use without mention.

Proposition 1.1 If X is a topological space and Y is a dense subspace of X, then for any A ⊆ Y, A is nowhere dense in Y if and only if A is nowhere dense in X. Similarly, A is meager in Y if and only if A is meager in X. □

2 Relationships between various Polish spaces

We begin by showing that for any two perfect Polish spaces X and Y, perfect(X) and perfect(Y) are equivalent statements.

Proposition 2.1 We have the following implications.

(a) Suppose X is a perfect Polish space and perfect(X) holds. Then perfect(ω^{ω}) holds.

(b) perfect(ω^{ω}) implies perfect(X) for every perfect Polish space X.

Proof. We will use the well-known fact that every perfect Polish space X has a dense G_{δ} subset Y homeomorphic to ω^{ω}. (To get Y, first remove the boundaries of the elements of a countable base for X. What remains is a zero-dimensional dense G_{δ}. Remove a countable dense subset of this dense G_{δ} and call the result Y. Then Y is a perfect Polish space which is zero-dimensional and has no compact open sets and hence is homeomorphic to ω^{ω}.)

(a) Let Y be a residual subspace of X homeomorphic to ω^{ω}. Let A be a nonmeager set in Y. In X, A is nonmeager so there is a nowhere dense perfect set C so that A is nonmeager in C. By replacing C by the closure of one of its nonempty open subsets, we may assume that A is everywhere nonmeager in C. In particular, A ∩ C is dense in C. Note that F = Y ∩ C is closed relative to Y, is nonempty and has no isolated points (because it contains A ∩ C which is dense in C). Since F is dense in C, A ∩ C = (A ∩ Y) ∩ C = A ∩ F is not meager in F. Also, because Y is dense in X and F is nowhere dense in X, F is also nowhere dense in Y.

(b) Let X be a perfect Polish space. Let Y be a residual subspace of X homeomorphic to ω^{ω}. Let A ⊆ X be nonmeager. Then A ∩ Y is nonmeager in X and hence in Y as well since Y is dense. By perfect(ω^{ω}), there is a nowhere dense perfect set C in Y such that A ∩ C is nonmeager in C. If P denotes the closure of C in X, then, since C is dense in P, A ∩ C is nonmeager in P and hence A ∩ P is also nonmeager in P. P is perfect since it is the closure of a nonempty set without isolated points. P is nowhere dense since it is the closure of a set which is nowhere dense in Y and hence in X as well. □

Part (b) holds for cantor(·) by an easier argument.

Proposition 2.2 cantor(ω^{ω}) implies cantor(X) for every perfect Polish space X.

Proof. Similar to the proof of Proposition 2.1(b), except that this time the proof yields a nowhere dense Cantor set C ⊆ Y such that A ∩ C is nonmeager in C and then we are done. □

We do not know whether (a) holds for cantor(·).

Problem 2.3 Does cantor(2^{ω}) imply cantor(ω^{ω})?

Problem 2.4 Does cantor([0, 1]) imply cantor([0, 1] × [0, 1])?

Of course, cantor([0, 1]) ≡ perfect([0, 1]) ≡ perfect(2^{ω}) ≡ cantor(2^{ω}), so these two questions have equivalent hypotheses.

We introduce one more version of perfect(X) based on the following observation. Suppose that perfect(ω^{ω}) holds. Then for any nonmeager set A, we have a nowhere dense perfect set P such that A ∩ P is nonmeager in P. Replacing P by the closure of one of its open sets, we may assume that A ∩ P is everywhere nonmeager in P. Then if P has a compact open subset U, then U is a Cantor set and A ∩ U is nonmeager in U. Otherwise, P itself is homeomorphic to ω^{ω}. Hence, the perfect set P in the conclusion of perfect(ω^{ω}) can always be taken to be either a closed nowhere dense copy of ω^{ω} or a Cantor set. Let baire(X) be the strengthening of perfect(X) in which we require that the perfect nowhere dense sets in the definition be homeomorphic to the Baire space ω^{ω}. Of course a Polish space need not contain any closed copies of ω^{ω}, so baire(X) can fail. However, when X = ω^{ω} it would seem reasonable that baire(X) might hold, and we will show in the next section that baire(ω^{ω}) is indeed consistent. Its relationship to cantor(ω^{ω}) is unclear to us.

Problem 2.5 (a) Does perfect(ω^{ω}) imply that one of baire(ω^{ω}) or cantor(ω^{ω}) must hold? (b) Does either of baire(ω^{ω}) or cantor(ω^{ω}) imply the other?

3 Consistency results

We now turn to the proof of the consistency of cantor(ω^{ω}) and baire(ω^{ω}). We need a variation on the following result which forms part of the proof of [Sh1980, Theorem 4.7] which states that if ZFC is consistent, then so is ZFC + 2^{ω} = ω_{2} + “There is a universal (linear) order of power ω_{1}.”

Theorem 3.1 If ZFC is consistent, then so is ZFC + both of the following statements.

(a) There is a nonmeager set in ℝ of cardinality ω_{1}.

(b) Let A and B be everywhere nonmeager subsets of ℝ of cardinality ω_{1}. Then A and B are order-isomorphic.

We shall need the following variant of this result.

Theorem 3.2 If ZFC is consistent, then so is ZFC + both of the following statements.

(a)′ Every nonmeager set in ℝ has a nonmeager subset of cardinality ω_{1}.

(b)′ Let A and B be everywhere nonmeager subsets of ℝ of cardinality ω_{1}. Suppose we are given countable dense subsets A_{0} ⊆ A and B_{0} ⊆ B. Then A and B are order-isomorphic by an order isomorphism taking A_{0} isomorphically to B_{0}.

Problem 3.3 In the presence of (a), does (b) imply (b)′?

We shall in fact verify in Theorem 4.9 that in (b)′ we can even ask that given pairwise disjoint countable dense subsets A_{i}, i < ω, of A and pairwise disjoint countable dense subsets B_{i}, i < ω, of B, the order-isomorphism of A and B takes A_{i} isomorphically to B_{i} for each i < ω. As explained in the introduction, the proof is similar to the one in [Sh1980], but as the proof is quite technical and the argument in [Sh1980] is only a brief sketch, we need to give the argument in some detail in order to be clear. We do that in the next section. Here, we derive the consequences of interest to us for this paper. The definition of baire(X) is given at the end of Section 2.

Theorem 3.4 Assume (a)′ and (b)′. Then cantor(ω^{ω}) and baire(ω^{ω}) both hold.

Proof. We will use the following elementary fact.

Fact 3.5 If K, L ⊆ ℝ are dense and h: K → L is an order isomorphism, then h extends to an order isomorphism of ℝ. □

Suppose that A ⊆ ℝ \ ℚ is not meager. We wish to find a Cantor set C ⊆ ℝ \ ℚ such that A ∩ C is nonmeager relative to C. By (a)′, we may assume that A has cardinality exactly ω_{1}. A is everywhere nonmeager in some open interval (a, b). Let C ⊆ (a, b)\ℚ be a Cantor set, and, by (a)′, let B ⊆ C be a set of cardinality ω_{1} which is nonmeager relative to C. Then (A∪ℚ)∩(a, b) and (A∪B ∪ℚ)∩(a, b) are both everywhere nonmeager in (a, b) and both have cardinality ω_{1}. By (b)′, there is an orderisomorphism h: (A ∪ ℚ) ∩ (a, b) → (A ∪ B ∪ ℚ) ∩ (a, b) such that h[ℚ ∩ (a, b)] = ℚ ∩ (a, b). Extend h to (a, b) and denote the extension also by h. Since h is a homeomorphism, h^{−1}[C] is a Cantor set and h^{−1}[B] is non meager relative to h^{−1}[C]. Since h^{−1}[C] ⊆ ℝ \ ℚ and h^{−1}[B] ⊆ A, we are done.

To get baire(ω^{ω}), we make a different choice of C in the preceding argument. This time, choose C to be any Cantor set so that C ∩ ℚ is dense in C. Then h^{−1}[C] will have the same property, so h^{−1}[C \ ℚ] = h^{−1}[C] \ ℚ is closed nowhere dense in ℝ \ ℚ and homeomorphic to ω^{ω}. □

Remark 3.6 The reader can easily verify that a similar but simpler argument yields that (a)′ and (b) imply perfect(ℝ). An alternative proof of the consistency of perfect(ℝ) can by had by using Theorem 2 of [CS] which states that the following statement is consistent relative to ZFC:

For every A ⊆ 2^{ω} ×2^{ω} for which the sets A and A^{c} = (2^{ω} ×2^{ω} \A) are nowhere meager in 2^{ω} ×2^{ω} there is a homeomorphism f : 2^{ω} → 2^{ω} such that the set {x ∈ 2^{ω} : (x, f(x)) ∈ A} does not have the Baire property in 2^{ω}.

(A set has the Baire property if it has the form U△M where U is open and M is meager.) Note that the map 2^{ω} → f given by x ↦ (x, f(x)) is a homeomorphism. Hence the conclusion could be stated as “f ∩ A does not have the Baire property in f”. Since 2^{ω} × 2^{ω} is homeomorphic to 2^{ω} and the graph of a homeomorphism of 2^{ω} is a perfect nowhere dense set in 2^{ω} × 2^{ω}, the statement above implies the following special case of perfect(2^{ω}) (which is equivalent to perfect(ℝ)).

For every A ⊆ 2^{ω} for which the sets A and A^{c} are both nowhere meager in 2^{ω}, there is a perfect nowhere dense set P such that the set A ∩ P is not meager in P.

To reduce perfect(2^{ω}) to this special case, consider a nonmeager set A ⊆ 2^{ω}. A is everywhere nonmeager in some clopen set U. If A is comeager in some clopen set, then it contains a nowhere dense perfect set and we are done. Hence we may assume that, relative to U, A and A^{c} are both everywhere nonmeager. The clopen set U is homeomorphic to 2^{ω}, so we now find ourselves in the special case described above.

4 Order-isomorphisms of everywhere nonmeager sets

We now turn to the proof of the consistency of (a)′ and (b)′. We begin by recalling the basic properties of oracle-cc forcing. See [Sh1998, Chapter IV] for the details. A version of this material is also explained in [Bu, Sections 4–6].

Definition 4.1 A sequence

\overline{M} = ⟨M_{δ} : δ is a limit ordinal < ω_{1}⟩

is called an oracle if each M_{δ} is a countable transitive model of a sufficiently large fragment of ZFC, δ ∈ M_{δ} and for each A ⊆ ω_{1}, {δ : A ∩ δ ∈ M_{δ}} is stationary in ω_{1}.

The meaning of “sufficiently large” depends on the context. In a particular proof, some fragment of ZFC for which models can be produced in ZFC must suffice for all the oracles in the proof. The existence of an oracle is equivalent to ♢, (see [Ku, Theorem II 7.14]) and hence implies CH. We limit the definition of the \overline{M}-chain condition to partial orders of cardinality ω_{1}. This covers our present needs.

Associated with an oracle \overline{M}, there is a filter Trap(\overline{M}) generated by the sets

{δ < ω_{1} : δ is a limit ordinal and A ∩ δ ∈ M_{δ}}, A ⊆ ω_{1}.

This is a proper normal filter containing all closed unbounded sets.

Definition 4.2 If P is any partial order, P′ ⊆ P, and 𝔇 is any class of sets, then we write P′ <_{𝔇} P to mean that every predense subset of P′ which belongs to 𝔇 is predense in P.

Definition 4.3 A partial order P satisfies the \overline{M}-chain condition, or simply is \overline{M}-cc, if there is a one-to-one function f: P → ω_{1} such that

{δ < ω_{1} : δ is a limit ordinal and f^{−1}(δ) <_{M_{δ,f}} P }

belongs to Trap(\overline{M}), where M_{δ,f} = \{f^{−1}(A) : A ⊆ δ, A ∈ M_{δ}\}.

It is not hard to verify that if P is \overline{M}-cc, then P is ccc. Also, any one-to-one function g: P → ω_{1} can replace f in the definition.

Proposition 4.4 The \overline{M}-cc satisfies the following properties.

(1) If α < ω_{2} is a limit ordinal, ⟨⟨P_{β}⟩_{β≤α}, ⟨Q̇_{β}⟩_{β<α}⟩ is a finite-support α-stage iteration of partial orders, and for each β < α, P_{β} is \overline{M}-cc, then P_{α} is \overline{M}-cc.

(2) If P is \overline{M}-cc, then there is a P -name \overline{M}^{∗} for an oracle such that for each P -name Q̇ for a partial order, if ⊩_{P} “Q̇ is \overline{M}^{∗}-cc” then P ∗ Q̇ is \overline{M}-cc.

(3) If \overline{M}_{α}, α < ω_{1}, are oracles, then there is an oracle \overline{M} such that for any partial order P, if P is \overline{M}-cc, then P is \overline{M}_{α}-cc for all α < ω_{1}.

We will need the following lemmas.

Lemma 4.5 Let \overline{M} = ⟨M_{δ} : δ < ω_{1}⟩ be an oracle and let A and B be everywhere nonmeager subsets of ℝ. Suppose we are given pairwise disjoint countable dense subsets A_{i}, i < ω, of A and pairwise disjoint countable dense subsets B_{i}, i < ω, of B. Then there is a forcing notion P satisfying the \overline{M}-cc such that for every G ⊆ P generic over V, V [G] ⊧ A and B are order-isomorphic by an order isomorphism taking A_{i} isomorphically to B_{i} for each i < ω.

Proof. Fix well-orderings of A and B in type ω_{1}. (CH holds because there is an oracle.) We will inductively define one-to-one enumerations ⟨a_{α} : α < ω_{1}⟩ of A and ⟨b_{α} : α < ω_{1}⟩ of B and functions f_{δ}, δ < ω_{1}. We let A_{δ} = {a_{α} : ωδ ≤ α < ω(δ + 1)} and B_{δ} = {b_{α} : ωδ ≤ α < ω(δ + 1)} for δ < ω_{1}. For A′ ⊆ A and B′ ⊆ B, let P (A′, B′) denote the set of finite partial order-preserving maps p: A′ → B′ such that p[A_{δ}] ⊆ B_{δ} for all δ < ω_{1}. We also use the notation

A ↾ α = {a_{β} : β < α}, B ↾ α = {b_{β} : β < α}.

We will arrange that the following conditions hold.

(1) The sets A_{δ} and B_{δ} are dense in ℝ.

(2) For δ < ω, the sets A_{δ} and B_{δ} are as in the hypothesis.

(3) For each δ < ω_{1}, f_{δ} is a bijective map of P (A ↾ ωδ, B ↾ ωδ) onto ωδ.

(4) For each δ < δ′ < ω_{1}, f_{δ} ⊆ f_{δ′}.

(5) For each infinite δ < ω_{1}, the predense subsets of P (A ↾ ωδ, B ↾ ωδ) which have the form f_{δ}^{−1}[S] for some S ∈ ⋃_{η≤δ} M_{η} remain predense in P (A ↾ ω(δ + 1), B ↾ ω(δ + 1)).

To do this, we proceed as follows. The construction of the functions f_{δ} is dictated by (4) at limit stages, and f_{δ+1} is an arbitrary extension of f_{δ} satisfying (3). The elements of A_{δ} and B_{δ} for δ < ω are given by (2). For δ ≥ ω, by induction on δ we choose the elements of A_{δ} and B_{δ} by alternately defining a_{ωδ+n} and b_{ωδ+n}, beginning with a_{ωδ} when δ is even and with b_{ωδ} when δ is odd. Let us illustrate the construction with the case where δ is even. Fix an enumeration ⟨I_{m} : 0 < m < ω⟩ of the nonempty open intervals with rational endpoints. The first element a_{ωδ} is simply the least element, in the well-ordering of A fixed at the beginning of the proof, which is different from any of the elements of A chosen so far. We now choose b_{ωδ}, a_{ωδ+1}, b_{ωδ+1}, a_{ωδ+2}, b_{ωδ+2}, … in that order. For n > 0, we pick a_{ωδ+n} and b_{ωδ+n} from I_{n} to ensure A_{δ} and B_{δ} will be dense.

To choose one of these elements, say b_{ωδ+n}, let N be a countable elementary submodel of H_{θ}, for a suitably large θ, such that A, B, f_{δ}, the sequences ⟨a_{α} : α ≤ ωδ + n⟩ and ⟨b_{α} : α < ωδ + n⟩, and ⋃_{η≤δ} M_{η} are all elements of N. Choose b_{ωδ+n} to be a member of B which is a Cohen real over N.

We must check that the construction gives (5). Let D be a predense subset of P (A ↾ ωδ, B ↾ ωδ) of the appropriate form. In particular, we have D ∈ N. We will show that D remains predense in P (A ↾ ωδ + n + 1, B ↾ ωδ + n + 1).

Remark 4.6 We are showing by induction on n that D remains predense in P (A ↾ ωδ + n + 1, B ↾ ωδ + n) and then in P (A ↾ ωδ + n + 1, B ↾ ωδ + n + 1). (This establishes (5) since each member of P (A ↾ ω(δ + 1), B ↾ ω(δ + 1)) belongs to P (A ↾ ωδ + n, B ↾ ωδ + n) for some n < ω.) Our current stage has the second form. Note that at the stage n = 0, we first consider the passage from P (A ↾ ωδ, B ↾ ωδ) to P (A ↾ ωδ + 1, B ↾ ωδ). But these two partial orders are equal because there is no legal target value for a_{ωδ} until b_{ωδ} is chosen. So the preservation of the predense sets trivially holds at that stage. In particular, it does not matter that a_{δω} is not Cohen generic over the previous construction.

Let

p ∈ P (A ↾ ωδ + n + 1, B ↾ ωδ + n + 1) \ P (A ↾ ωδ + n + 1, B ↾ ωδ + n).

Then p has the form q ∪ {(a, b_{ωδ+n})} for some q ∈ P (A ↾ ωδ + n + 1, B ↾ ωδ + n) and a ∈ {a_{ωδ+m} : m ≤ n}. Fix r ∈ D. The set

{b ∈ ℝ : q ∪ {(a, b)} is compatible with r} ∈ N

(“compatible with” here means only that q ∪ {(a, b)} ∪ r is a finite order isomorphism) is open and hence its complement C_{r} is closed, as is the set C_{D} = ⋂_{r∈D} C_{r} of b for which q ∪ {(a, b)} is incompatible with every member of D. Since p is an partial order isomorphism, there are open rational intervals J_{1} and J_{2} such that J_{1} ∩ dom p = {a}, J_{2} ∩ ran p = {b_{ωδ+n}}. Note that whenever x ∈ J_{1} and b ∈ J_{2}, q ∪ {(x, b)} is a partial isomorphism.

Claim 4.7 C_{D} is nowhere dense in J_{2}.

Proof. Fix a nonempty open subinterval J of J_{2}. There is an extension of q by members of A_{0} × B_{0}—the point of using A_{0} and B_{0} being simply that they are dense and contained in A(ωδ + n + 1) and B(ωδ + n), respectively—which adds two points in J_{1} × J straddling the line x = a. So this extension has the form

q′ = q ∪ {(x_{1}, y_{1}), (x_{2}, y_{2})}, x_{1} < a < x_{2}, y_{1} < y_{2}

where (x_{1}, x_{2}) ⊆ J_{1} and (y_{1}, y_{2}) ⊆ J. Since q′ ∈ P (A ↾ ωδ + n + 1, B ↾ ωδ + n), by the induction hypothesis D must have an element r compatible with this extension. Since a ∉ A(ωδ), we have a ∉ dom r. Let x′_{1}, x′_{2} be the closest members of dom(q′ ∪ r) to the left and right of a, respectively. Write y′_{1} = r(x′_{1}), y′_{2} = r(x′_{2}). Then (y′_{1}, y′_{2}) ⊆ (y_{1}, y_{2}) ⊆ J and for any choice of b ∈ (y′_{1}, y′_{2}), q ∪ {(a, b)} is compatible with r. Hence (y′_{1}, y′_{2}) is disjoint from C_{r} and hence from C_{D}. This proves the claim. □

Thus, b_{ωδ+n} ∉ C_{D} and hence p is compatible with some member of D. This establishes (5).

Now take P = P (A, B). The fact that P forces the desired order-isomorphism of A and B is clear from (1) and (2). To see that P is \overline{M}-cc, let f = ⋃_{δ<ω_{1}} f_{δ}: P → ω_{1}. For any δ < ω_{1} we have f^{−1}[ωδ] = P (A(ωδ), B(ωδ)) and for each S ⊆ ωδ whenever a set D of the form f^{−1}[S] = f_{δ}^{−1}[S] belongs to M_{δ} and is predense in P (A(ωδ), B(ωδ)), a simple induction on δ′ using (5) shows that if δ is infinite and δ < δ′ ≤ ω_{1}, then D is predense in P (A(ωδ′), B(ωδ′)). In particular, D is predense in P = P (A(ω_{1}), B(ω_{1})). For a club of δ < ω_{1} we have ωδ = δ, so this shows that P satisfies the \overline{M}-cc. □

Lemma 4.8 Assume ♢. Let A be a nonmeager subset of ℝ. Then there is an oracle \overline{M} = ⟨M_{δ} : δ < ω_{1}⟩ such that if P is any partial order satisfying the \overline{M}-cc, then ⊩_{P} “A is nonmeager”.

Proof. This is [Sh1998, Example IV 2.2]. □

Theorem 4.9 If ZFC is consistent, then so is ZFC + both of the following statements.

(a) Every nonmeager set in ℝ has a nonmeager subset of cardinality ω_{1}.

(b) Let A and B be everywhere nonmeager subsets of ℝ of cardinality ω_{1}. Suppose we are given pairwise disjoint countable dense subsets A_{i}, i < ω, of A and pairwise disjoint countable dense subsets B_{i}, i < ω, of B. Then A and B are order-isomorphic by an order isomorphism taking A_{i} isomorphically to B_{i} for each i < ω.

Proof. Start with a ground model of V = L. Fix a diamond sequence

⟨(f_{α}, g_{α}, h_{α}) : α < ω_{2}, cof(α) = ω_{1}⟩

for trapping triples (f, g, h) consisting of:

(1) A function f: ω_{2} → ([ω_{2}]^{≤ω})^{ω}. The idea of f is that, with ω_{2} identified with the ccc partial order we are about to build, [ω_{2}]^{≤ω} contains the maximal antichains. Thus, ([ω_{2}]^{≤ω})^{ω} contains a name for each real number (construed as a subset of ω). Then for any nonmeager set X in the extension, we can find a ground model function f: ω_{2} → ([ω_{2}]^{≤ω})^{ω} enumerating the names of the elements of X.

(2) Functions g, h: ω_{1} → ([ω_{2}]^{≤ω})^{ω} intended to represent (enumerations of the names for the elements of) everywhere nonmeager sets of cardinality ω_{1} with each of the sets {g(ωi + n) : n < ω} and {h(ωi + n) : n < ω}, for i < ω, dense in ℝ.

So for each α < ω_{2} of cofinality ω_{1}, f_{α}: α → ([α]^{≤ω})^{ω}, and g_{α}, h_{α}: ω_{1} → ([α]^{≤ω})^{ω}. Also, for each (f, g, h) as in (1)–(2), {α < ω_{2} : cof(α) = ω_{1}, f ↾ α = f_{α}, g ↾ α = g_{α} and h ↾ α = h_{α}} is stationary in ω_{2}.

We will inductively define an ω_{2}-stage finite support iteration

⟨⟨P_{α}⟩_{α≤ω_{2}}, ⟨Q̇_{α}⟩_{α<ω_{1}}⟩

as well as a P_{α}-names \overline{M}_{α} for oracles and one-to-one functions F_{α}: P_{α} → ω_{2} for α < ω_{2} such that the range of each F_{α} is an initial segment of ω_{2} which includes α and for β < α < ω_{2}, we have F_{β} ⊆ F_{α}. (At each stage, F_{α} is any function satisfying these conditions.)

For α < ω_{2}, we will let Ẋ_{α} denote the P_{α}-name for the set of real numbers whose elements have the names

⋃n<ω{n} × F_{α}^{−1}(f_{α}(ξ)(n)), ξ < α.

Similarly, we will let Ȧ_{α} and Ḃ_{α} denote the ω_{1}-sequences of P_{α}-names for real numbers

⟨_{⋃n<ω}{n} × F_{α}^{−1}(g_{α}(ξ)(n)) : ξ < ω_{1}⟩

and

⟨_{⋃n<ω}{n} × F_{α}^{−1}(h_{α}(ξ)(n)) : ξ < ω_{1}⟩

respectively. At stage α < ω_{2} of the construction, if cof(α) = ω_{1} and if

⊩_{P_{α}} Ẋ_{α} is not meager,

then we use Lemma 4.8 to get a P_{α}-name \overline{M}′_{α} for an oracle so that if P is any forcing notion which satisfies the \overline{M}′_{α}-cc, then X_{α} remains nonmeager after forcing with P. Otherwise, in particular if cof(α) ≠ ω_{1}, we let \overline{M}′_{α} be any P_{α}-name for an oracle.

For β < α, let P_{βα} be the usual P_{β}-name for a partial order such that P_{α} is isomorphic to a dense subset of P_{β} ∗ P_{βα} (see [Ba]). Let \overline{M}_{βα} be a P_{α}-name for an oracle such that

If ⊩_{P_{β}} “P_{β,α} is \overline{M}_{β}-cc and ⊩_{P_{β,α}} Q̇_{α} is \overline{M}_{βα}-cc”,

(1)

then ⊩_{P_{β}} “P_{β,α+1} = P_{β,α} ∗ Q̇_{α} is \overline{M}_{β}-cc”.

(There is such an \overline{M}_{βα} by Proposition 4.4(2). In (1), \overline{M}_{βα} is actually a P_{β}-name for a P_{β,α}-name for an oracle. We denote the corresponding P_{α}-name also by \overline{M}_{βα}.)

Let \overline{M}_{α} be a P_{α}-name for an oracle such that

(2)  ⊩_{P_{α}} “If Q̇_{α} is \overline{M}_{α}-cc, then Q̇_{α} is \overline{M}′_{α}-cc and \overline{M}_{βα}-cc for all β < α”.

(Use Proposition 4.4(3).)

Now, if cof(α) = ω_{1} and if

⊩_{P_{α}} The ranges of Ȧ_{α}, Ḃ_{α} are everywhere nonmeager and each of the sets

{Ȧ_{α}(ωi + n) : n < ω}, {Ḃ_{α}(ωi + n) : n < ω}, for i < ω, is dense in ℝ.

then use Lemma 4.5 to get a P_{α}-name Q̇_{α} for a partial order satisfying the \overline{M}_{α}-cc and forcing an isomorphism between A_{α} and B_{α} as described in the statement of the lemma. In all other cases, take Q̇_{α} to name the partial order Q for adding one Cohen real. We have thus

(3)  ⊩_{P_{α}} “Q̇_{α} satisfies the \overline{M}_{α}-cc”.

Now suppose that for some P_{ω_{2}}-name Ẋ we have

⊩_{P_{ω_{2}}} Ẋ is not meager.

(Every nonmeager set in any extension has a name forced by the weakest condition to be nonmeager since there is always a nonmeager set.) Fix a name ḟ such that

⊩_{P_{ω_{2}}} ḟ: ω_{2} → Ẋ is onto.

Then define f: ω_{2} → ([ω_{2}]^{≤ω})^{ω} so that if

τ_{ξ} = _{⋃n<ω}{n} × F^{−1}(f(ξ)(n)), ξ < ω_{2},

then for each ξ < ω_{2},

⊩_{P_{ω_{2}}} ḟ(ξ) = τ_{ξ}.

There is a closed unbounded set C ⊆ ω_{2} such that for each α ∈ C of cofinality ω_{1} we have:

(i) f ↾ α: α → ([α]^{≤ω})^{ω}.

(ii) ∀ξ < α, τ_{ξ} is a P_{α}-name.

(iii) ⊩_{P_{α}} {τ_{ξ} : ξ < α} is not meager.

(For (iii), note that when α has cofinality ω_{1}, each P_{α}-name for a meager set is a P_{β}-name for some β < α. Thus, if M is an elementary submodel of H_{θ} for a suitably large θ such that |M| = ω_{1}, M^{ω} ⊆ M, ⟨τ_{ξ} : ξ < ω_{2}⟩ ∈ M and α = M ∩ ω_{2} ∈ ω_{2} has cofinality ω_{1}, then for each (nice) P_{α}-name σ for a meager Borel set set, we have σ ∈ M and hence M knows about a maximal antichain of conditions each deciding a ξ for which τ_{ξ} is forced not to be in σ. The antichain is countable and hence contained in M. For each condition in the antichain, the least ξ which it decides is in M and hence below α. Hence ⊩_{P_{α}} “{τ_{ξ} : ξ < α} is not contained in σ”.)

Choose such an α of cofinality ω_{1} for which f ↾ α = f_{α}. By (i) and (ii), the definition of τ_{ξ} would not change if we used f_{α} instead of f and F_{α} instead of F. Then from the definition of Ẋ_{α} we get

⊩_{P_{α}} Ẋ_{α} = {τ_{ξ} : ξ < α}.

So at stage α we chose a P_{α}-name \overline{M}_{α} and we arranged that

⊩_{P_{α}} “P_{α,γ} is \overline{M}_{α}-cc”.

(This follows easily by induction on γ ≥ α and Propositions 4.4(1,2). (Recall that P_{α,γ} can be viewed in canonical way as an iteration: see [Ba]. At limits γ use Propositions 4.4(1). At stages γ + 1, use (3) to get ⊩_{P_{γ}} “Q̇_{γ} satisfies the \overline{M}_{γ}-cc” and then use (2) and (1) with (β, α) replaced by (α, γ).)

Hence, by the choice of \overline{M}_{α},

(4)  ⊩_{P_{α}}⊩_{P_{α,γ}} Ẋ_{α} is not meager

from which it follows that

⊩_{P_{α}}⊩_{P_{α,ω_{2}}} Ẋ_{α} is not meager

since if this failed then we would have

p ⊩_{P_{α}} q ⊩_{P_{α,ω_{2}}} Ẋ_{α} ⊆ Ḃ

for some conditions p ∈ P_{α}, q ∈ P_{α,ω_{2}} and some name Ḃ for a meager Borel set. But then for some γ, we have α < γ < ω_{2}, q ∈ P_{α,γ} and Ḃ is a P_{γ}-name and this contradicts (4).

By what we have established, there are guaranteed to be sets of cardinality ω_{1} which are not meager in any extension by P_{ω_{2}}. Hence there are guaranteed to be everywhere nonmeager sets of cardinality ω_{1}. Suppose that for some P_{ω_{2}}-names Ȧ and Ḃ for ω_{1}-sequences we have

⊩_{P_{ω_{2}}} The ranges of Ȧ, Ḃ are everywhere nonmeager and each of the sets

{Ȧ(ωi + n) : n < ω}, {Ḃ(ωi + n) : n < ω}, for i < ω, is dense in ℝ.

(By what we just said, every pair of everywhere nonmeager sets A and B of cardinality ω_{1}, together with choices of countably many disjoint countable dense subsets of each one, has a name such that the weakest condition forces the desired properties.)

Define g, h: ω_{1} → ([ω_{2}]^{≤ω})^{ω} so that if

σ_{ξ} = _{⋃n<ω}{n} × F^{−1}(g(ξ)(n)), ξ < ω_{1}

and

τ_{ξ} = _{⋃n<ω}{n} × F^{−1}(h(ξ)(n)), ξ < ω_{1}

then for each ξ < ω_{1},

⊩_{P_{ω_{2}}} Ȧ(ξ) = σ_{ξ}

and

⊩_{P_{ω_{2}}} Ḃ(ξ) = τ_{ξ}.

For all large enough α < ω_{2}, we have:

(i) g, h: ω_{1} → ([α]^{≤ω})^{ω}.

(ii) ∀ξ < α, σ_{ξ} and τ_{ξ} are P_{α}-names.

Choose any such α of cofinality ω_{1}. By (i) and (ii), the definitions of σ_{ξ} and τ_{ξ} would not change if we used g_{α} instead of g, h_{α} instead of h, and F_{α} instead of F. Then from the definitions of Ȧ_{α} and Ḃ_{α} we get

⊩_{P_{α}} The ranges of Ȧ_{α}, Ḃ_{α} are everywhere nonmeager and each of the sets

{Ȧ_{α}(ωi + n) : n < ω}, {Ḃ_{α}(ωi + n) : n < ω}, for i < ω, is dense in ℝ.

(Being everywhere nonmeager is trivially downward absolute.) Then Q̇_{α} was chosen to add an order isomorphism between A_{α} and B_{α} of the desired type.

This completes the proof of the theorem. □

5 A measure-theoretic analog of perfect(2^{ω})

A measure theoretic version of Laczkovich’s question is not completely obvious because perfect sets carry many measures. We consider the following measures on 2^{ω} which we will call canonical. Given P ⊆ 2^{ω} a perfect set, define

T_{P} = {s ∈ 2^{<ω} : P ∩ U(s) ≠ ∅},

where U(s) = {x ∈ 2^{ω} : s ⊆ x}. We say that s ∈ T_{P} splits iff both s0 and s1 are in T_{P}. The canonical measure μ_{P} is the one supported by P and determined by declaring μ_{P} (U(s)) = 1/2^{n} iff s ∈ T_{P} and |{i < |s| : s ↾ i splits}| = n. An equivalent view is to take the natural map from 2^{<ω} to the splitting nodes of T_{P} and the homeomorphism h : 2^{ω} → P ⊆ 2^{ω} induced by it and then μ_{P} is the measure corresponding to the product measure μ on 2^{ω}, i.e., μ_{P} (A) = μ(h^{−1}(A)).

Theorem 5.1 It is relatively consistent with ZFC that for any set B ⊆ 2^{ω} which is not of measure zero, there exists a perfect set P of measure zero such that B ∩ P does not have measure zero in the canonical measure μ_{P} on P.

Proof. The model is the one used by Rosłanowski and Shelah in the proof of [RS, Theorem 3.2]. It is obtained by forcing over a model of CH with an ω_{2}-stage countable support iteration ⟨⟨ℙ_{α}⟩_{α≤ω_{2}}, ⟨ℚ̇_{α}⟩_{α<ω_{2}}⟩ of the measured creature forcing ℚ = ℚ_{4}^{mt}(K^{∗}, Σ^{∗}, F^{∗}) defined in [RS, Section 2]. We use the notation of [RS] concerning this partial order. The definition involves in particular a rapidly growing sequence of powers of 2, ⟨N_{i} = 2^{M_{i}} : i < ω⟩. Forcing with ℚ gives rise to a continuous function h: ∏_{i<ω} N_{i} → 2^{ω}. We will make use of the following result concerning this function. The measure on ∏_{i<ω} N_{i} in this proposition is the product of the uniform probability measures on the factors and the measure on 2^{ω} is the usual product measure. In the remainder of this proof, we denote both of these measures, as well as their product, by μ, letting the context distinguish them.

[RS, Proposition 2.6] Suppose that A ⊆ ∏_{i<ω} N_{i} × 2^{ω} is a set of outer measure one.

Then, in V^{ℚ}, the set

{x ∈ ∏_{i<ω} N_{i} : (x, h(x)) ∈ A}

has outer measure one.

We shall also need to know that ℚ is proper and that countable support iterations of ℚ preserve Lebesgue outer measure. The former is [RS, Corollary 1.14]. The latter is explained in the proof of [RS, Theorem 3.2]. (The explanation refers the reader to some very general preservation theorems for iterated forcing. For the reader who wants to verify this without learning these general theorems, we indicate that it also follows from the special case of these theorems, preservation of ⊏^{random}, given in [Go] by imitating the proof in [Pa] that Laver forcing satisfies what is called there ⋆ and by noting that ⋆ implies preservation of ⊏^{random}.)

Recall that N_{i} = 2^{M_{i}}. We identify N_{i} with the set of binary sequences of length M_{i}. The map h : ∏_{i<ω} N_{i} → 2^{ω} is determined from a generic sequence of finite maps (W (i) : N_{i} → 2 : i < ω) added by ℚ. h is defined by h(x)(i) = W (i)(x(i)) for each i. We use the W (i)’s to define a perfect set P ⊆ 2^{ω} by the condition that x ∈ P if and only if there exists y ∈ 2^{ω} such that x is the concatenation of the sequence s_{0}, i_{0}, s_{1}, i_{1}, … where y is the concatenation of s_{0}, s_{1}, s_{2}, … and where each s_{k} has length M_{k} and i_{k} = W (k)(s_{k}) ∈ {0, 1}. P is essentially the same as the graph of h but we spell out the details to be sure the canonical measure is the one we want. Another way to define P is as follows:

(i) Let l_{i} = M_{i} + ∑_{j<i}(M_{j} + 1). Let l_{−1} = −1. The l_{i}, i < ω, are the nonsplitting levels of the tree T_{P} which can be determined by the next two conditions.

(ii) If s ∈ T_{P} and l_{i−1} < |s| < l_{i} then both s0 and s1 are in T_{P}.

(iii) If s ∈ T_{P} and |s| = l_{i}, then only sj in T_{P} where W (i)(t) = j and s = rt is the concatenation of r and t where |t| = M_{i} and r has the appropriate length.

(iv) Define

P = [T_{P}] =^{def} {x ∈ 2^{ω} : ∀n x ↾ n ∈ T_{P} }

Every time we pass a nonsplitting level l_{i} we lose half the measure and so P is a perfect set of measure zero for the usual measure on 2^{ω}.

Let ρ : ∏_{i<ω} N_{i} × 2^{ω} → 2^{ω} be the natural homeomorphism given by ρ(x, z) is the concatenation of the sequence x_{0}, z_{0}, x_{1}, z_{1}, … where we are identifying N_{i} with the set of binary sequences of length M_{i}.

Claim 5.2 ρ is measure-preserving.

Proof. By a standard uniqueness theorem for the extension of a measure from an algebra to the σ-algebra it generates, it suffices to verify that ρ^{−1}[C] has the same measure as C for every clopen set C ⊆ 2^{ω}. As above, for k < ω and s ∈ 2^{k}, let us write U(s) = {x ∈ 2^{ω} : s ⊆ x}. Similarly, for s ∈ ∏_{i<k} N_{i} we write V (s) = {x ∈ ∏_{i<ω} N_{i} : s ⊆ x}. Every clopen set C ⊆ 2^{ω} can be partitioned into clopen sets of the form U(r), where for some k < ω, s^{r} ∈ ∏_{i<k} N_{i} and t^{r} ∈ 2^{k}, r is the concatenation of s_{0}^{r}, t_{0}^{r}, …, s_{k−1}^{r}t_{k−1}^{r}. (These are simply the basic open sets U(r) for which r has length ∑_{i<k}(M_{i} + 1) for some k < ω.) Hence it suffices to verify μ(ρ^{−1}[U(r)]) = μ(U(r)) for r of this form. We have

μ(ρ^{−1}[U(r)]) = μ(V (s^{r}) × U(t^{r})) = (∏_{i<k} 2^{−M_{i}}) 2^{−k} = 2^{− ∑_{i<k}(M_{i}+1)} = μ(U(r)).

This proves the claim. □

Let g: ∏_{i<ω} N_{i} → ∏_{i<ω} N_{i} × 2^{ω} be the homeomorphism of ∏_{i<ω} N_{i} onto the graph of h given by g(x) = (x, h(x)). We have ρ[h] = P (i.e., the graph of h corresponds to P under ρ).

Claim 5.3 For any Borel set B ⊆ 2^{ω}

μ_{P} (B) = μ (g^{−1}[ρ^{−1}[B]])

and similarly for outer measure.

Proof. Since the range of g is the graph of h, we have

μ (g^{−1}[ρ^{−1}[B]]) = μ (g^{−1}[ρ^{−1}[B] ∩ h]) = μ (g^{−1}[ρ^{−1}[B ∩ P]]) .

Similarly, since μ_{P} concentrates on P, μ_{P} (B) = μ_{P} (B ∩ P). Hence, it suffices to prove the claim for Borel subsets of P.

Given s ∈ ∏_{i<k} N_{i}, define t^{s} ∈ 2^{k} by t_{i}^{s} = W (i)(s(i)) for all i < k, and write r^{s} for the concatenation of s_{0}, t_{0}^{s}, …, s_{k−1}, t_{k−1}^{s}. Using the notation for basic open sets from the proof of the previous claim, we have

μ_{P} (U(r^{s})) = 2^{− ∑_{i<k} M_{i}} = ∏_{i<k} 2^{−M_{i}}.

Also, ρ^{−1}[U(r^{s})] = V (s) × U(t^{s}) and g^{−1}[ρ^{−1}[U(r^{s})]] = g^{−1}[V (s) × U(t^{s})] = V (s), so

μ (g^{−1}[ρ^{−1}[U(r^{s})]]) = μ(V (s)) = ∏_{i<k} 2^{−M_{i}}.

Thus, the claim holds for basic open sets of the form U(r^{s}). Every clopen subset of P is partitioned by such sets, so the claim holds for all clopen sets, and hence, as in the proof of Claim 5.2, for all Borel sets. This proves the claim. □

Now we prove Theorem 5.1 in the case that B ⊆ 2^{ω} has outer measure one. It follows from the usual Lowenheim-Skolem arguments that if we let B_{α} = V^{ℙ_{α}} ∩ B then there will exist α < ω_{2} such that B_{α} ∈ V^{ℙ_{α}} and

V^{ℙ_{α}} ⊧ B_{α} has outer measure one.

Letting A = ρ^{−1}(B_{α}) (which has outer measure one by Claim 5.2) in [RS, Proposition 2.6] cited above and using Claim 5.3, we have that

V^{ℙ_{α+1}} ⊧ B_{α} has μ_{P} outer measure one.

Because ℚ is proper, the remainder ℙ_{ω_{2}}/ℙ_{α+1} of the forcing is isomorphic in V^{ℙ_{α+1}} to a countable support iteration of ℚ and hence preserves outer measure. It follows that in the final model V^{ℙ_{ω_{2}}}, B has μ_{P} outer measure one.

Now in the case that B has outer measure less than one, replace it by B′ = Q + B where Q is a countable dense subset of 2^{ω}. Then B′ has outer measure one, and so we know there exists a measure zero perfect P such that B′ has positive μ_{P} outer measure. Hence for some q ∈ Q we have that q + B has positive μ_{P} outer measure. But then, B has positive μ_{q+P} outer measure.

This completes the proof of the theorem. □

Problem 5.4 Is it relatively consistent with ZFC to have simultaneously both the category property, perfect(2^{ω}), and its measure theory analogue, Theorem 5.1?

References

[Ba] J.E. Baumgartner, Iterated forcing, pp. 1-59 in ‘Surveys in Set Theory’, edited by A.R.D. Mathias, LMS Lecture notes 87, Cambridge, Cambridge Univ. Press, 1983.

[CS] K. Ciesielski, S. Shelah, Category analogue of sup-measurability problem, J. Appl. Anal., 6 (2000) 159–172.

[Bu] M.R. Burke, Liftings for Lebesgue measure, in Set theory of the reals (Ramat Gan, 1991), Israel Math. Conf. Proc., 6 (1993) 119–150.

[Go] M. Goldstern, Tools for your forcing construction, Set theory of the reals (Ramat Gan, 1991), 305–360, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993.

[Ku] K. Kunen, Set Theory, North-Holland, 1983.

[Pa] J. Pawlikowski, Laver’s forcing and outer measure, Set theory (Boise, ID, 1992–1994), 71–76, Contemp. Math., 192, Amer. Math. Soc., Providence, RI, 1996.

[RS] A. Roslanowski, S. Shelah, Measured creatures, http://front.math.ucdavis.edu/math.LO/0010070

[Sh1980] S. Shelah, Independence results, J. Symbolic Logic, 45 (1980) 563–573.

[Sh1998] S. Shelah, Proper and improper forcing, 2nd ed., Springer-Verlag, Berlin, 1998.

Appendix.

This appendix is intended for the electronic version only. Here is our original proof obtained May 1997.

A note on second category sets

The purpose of this note is to answer the following question of M. Laczkovich.

Does there exist (in ZFC) a second category set that is relative first category in every nowhere dense perfect set?

We shall show that the answer is no. We assume that the reader is familiar with oracle chain condition constructions. The technique is explained in [S1, pp.114–133] and all the details of a typical proof are written out in [B, Sections 4,5,6]. We shall follow the notation and general setup of the proof in [B].

Theorem It is consistent with ZFC that every second category subset of the real line has a nowhere dense subset of cardinality ω_{1} which is relatively second category in its closure.

Proof Assume V = L. As in [B, Section 5], we use an oracle-cc iteration of length ω_{2}, and it will suffice to prove the following lemma.

Main Lemma Let \overline{M} be an ω_{1}-oracle and let A be an everywhere nonmeager subset of 2^{ω}. Then there is a forcing notion P satisfying the \overline{M}-cc and a P -name K̇ of a nowhere dense perfect set such that for every G ⊆ P × Q generic over V (where Q is Cohen forcing), there is no Borel set T in V [G] such that

(a) T ∩ K̇[G] is meager relative to K̇[G].

(b) A ∩ K̇[G] ⊆ T.

Proof of the main lemma

Define a partial order P = P (⟨a_{α} : α < β⟩) where β ≤ ω_{1}, a_{α} ∈ A as follows. Let S_{β} = {a_{α} : α < β} ∪ {x : x is eventually constant}. Then the conditions in P are the pairs p = (F_{p}, n_{p}) where F_{p} is a finite subset of S_{β} and n_{p} < ω. Ťhe order on P is: p ≤ q if and only if F_{p} ⊇ F_{q}, n_{p} ≥ n_{q}, and x ∈ F_{p} implies x ↾ n_{q} = y ↾ n_{q} for some y ∈ F_{q}. Ľet K̇ be a P -name for the closure of ⋃{F : for some n < ω, (F, n) ∈ G}. As in [B, Section 6], the main lemma will follow if we prove the following:

Main Claim Let P_{δ} = P (⟨a_{α} : α < δ⟩), δ < ω_{1} be given, as well as a countable M_{δ}, P_{δ} ∈ M_{δ}, a condition (p^{∗}, r^{∗}) ∈ P_{δ} × Q and a P_{δ} × Q-names τ_{n}, n < ω, for relatively nowhere dense closed subsets of K̇. Then we can find a_{δ} ∈ A such that, letting P_{δ+1} = P (⟨a_{α} : α ≤ δ⟩), the following conditions hold:

(A) Every predense subset of P_{δ} which belongs to M_{δ} is a predense subset of P_{δ+1}. ˇ (B) There is a condition (p′, r′) ∈ P_{δ+1} ×Q extending (p^{∗}, r^{∗}) such that for all n < ω, (p′, r′) ⊩_{P_{δ+1}×Q} “a_{δ} ∉ τ_{n}”.

Proof of the main claim

Choose a sufficiently large regular λ and choose a countable N ≺ H_{λ} such that ρ, P_{δ}, ⟨a_{α} : α < δ⟩, ⟨τ_{n} : n < ω⟩, M_{δ} ∈ N. Choose a Cohen real over N, x = a_{δ} ∈ A, such that x ↾ n_{p^{∗}} = y^{∗} ↾ n_{p^{∗}} for some y^{∗} ∈ F_{p^{∗}}.

Proof of condition (A) Let J ⊆ P_{δ} be predense, J ∈ M_{δ}. We must show that J is predense in P_{δ+1}. Let p ∈ P_{δ+1}, p ∉ P_{δ}. By the definition of P_{δ+1}, p = p(x) = (F ∪ {x}, n) where x = a_{δ} and F ⊆ S_{δ}. Think of x as being generic for 2^{<ω} ordered the usual way. If p(x) is not compatible with any element of J, then some s ∈ 2^{<ω} forces this over N. We may assume n ≥ |s|. Since S_{δ} is dense, there is y ∈ S_{δ} such that y ↾ n = x ↾ n. Now choose p_{1} ∈ P_{δ}, a common extension of (F ∪ {y}, n) and some p_{2} ∈ J. If x is any Cohen real such that x ↾ n_{p_{1}} = y ↾ n_{p_{1}}, then clearly p(x) is compatible with p_{2}, contradiction.

Proof of condition (B) Let p′ = (F, n) where F = F_{p^{∗}} ∪ {x} and n = n_{p^{∗}}, and let r′ = r^{∗}. If N[x] believes that these satisfy (B) then they do and we are done. Otherwise, choose (p′′, r′′) in N[x] extending (p′, r′) and n < ω such that

N[x] ⊧ (p′′, r′′) ⊩_{P_{δ+1}×Q} “x ∈ τ_{n}”.

For some Cohen condition s ∈ 2^{<ω} we have

N ⊧ s ⊩ “p′′ = (F′′ ∪ {x}, n′′) and (p′′, r′′) ⊩_{P_{δ+1}×Q} x ∈ τ_{n}”,

where F′′ ⊆ S_{δ} contains a member u of [s] and without loss of generality n′′ ≥ |s|. Extend ((F′′, n′′), r′′) to ((F′′′, n′′′), r′′′) in P_{δ} × Q to decide a t ∈ 2^{<ω} such that u ↾ n′′ ⊆ t, [t] ∩ K̇ ≠ ∅ and [t] ∩ τ_{n} = ∅. By increasing n′′′ and/or t, we may arrange that n′′′ = |t|. Clearly [t] ∩ F′′′ ≠ ∅, so N models the following for p′′′ = (F′′′ ∪ {x}, n′′′):

N ⊧ t ⊩ “p′′′ ≤ (F′′′, n′′′), (F′′ ∪ {x}, n′′) and (p′′′, r′′′) ⊩_{P_{δ+1}×Q} x ∉ τ_{n} ”

which is a contradiction. This completes the proof of the main claim and of the theorem. □

References:

[B] Burke M.R., Liftings for Lebesgue measure, Israel Math. Conf. Proc., 6 (1993) 119–150.

[S1] Shelah S., Proper Forcing, Lecture Notes in Math. 940, Berlin, Springer-Verlag, 1982.