Abstract. We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.

166 (2000)

Covering of the null ideal may have countable cofinality

by

Saharon S h e l a h (Jerusalem and New Brunswick, NJ)

Abstract. We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.

0. Introduction. In the present paper we show that it is consistent that the covering of the null ideal has countable cofinality. Recall that the covering number of the null ideal (i.e. the ideal of measure zero sets) is defined as

cov(null) = min n

|P| : P ⊆ null and [

A∈P

A = R (= ^ω 2) o .

The question whether the cofinality of cov(null) is uncountable has been raised by D. Fremlin and has been around since the late seventies. It appears in Fremlin’s list of problems, [Fe94], as problem CO. Recall that for the ideal of meagre sets the answer is positive: A. Miller [Mi82] proved that the cofinality of the covering of category is uncountable. T. Bartoszy´ nski [Ba88] saw that b < ℵ ω is necessary (see [BaJu95, Ch. 5] for more results related to this problem). It should be noted that most people thought cf(cov(null)) = ℵ 0 was impossible.

The main result of this paper is the following:

Theorem 0.1. Con(cov(null) = ℵ ω + MA ℵ

) for each n < ω.

The presentation of the proof of 0.1 sacrifices generality for hopeful trans- parency. We finish by some further remarks, e.g. the exact cardinal assump- tion for 0.1. We try to make the paper self-contained for readers with basic knowledge of forcing.

2000 Mathematics Subject Classification: 03E17, 03E35.

Key words and phrases: null sets, cardinal invariants of the continuum, iterated forc- ing, ccc forcing.

Research supported by “The Israel Science Foundation” founded by The Israel Academy of Sciences and Humanities. Publication no. 592.

[109]

In a subsequent paper, [Sh 619], we deal with the question: “can every non-null set be partitioned into uncountably many non-null sets”, equiva- lently: “can the ideal of null sets which are subsets of a non-null subset of R be ℵ 1 -saturated”. P. Komj´ ath [Ko89] proved that it is consistent that there is a non-meagre set A such that the ideal of meagre subsets of A is ℵ 1 -saturated. The question whether a similar fact may hold for measure dates back to Ulam; see also Prikry’s thesis. It appears as question EL(a) on Fremlin’s list. In [Sh 619] we prove the following:

Theorem 1. It is consistent that there is a non-null set A ⊆ R such that the ideal of null subsets of A is ℵ 1 -saturated (of course, provided that “ZFC + ∃ measurable” is consistent ).

In [Sh 619] we also prove the following.

Theorem 2. It is consistent that :

(⊕) there is a non-null A ⊆ R such that for every f : A → R, the function f as a subset of the plane R × R is null

provided that “ZFC + there is a measurable cardinal” is consistent.

Notation 0.2. We denote:

• natural numbers by k, l, m, n and also i, j,

• ordinals by α, β, γ, δ, ζ, ξ (δ always limit),

• cardinals by λ, κ, χ, µ,

• reals by a, b and positive reals (normally small) by ε,

• subsets of ω or ^ω≥ 2 or Ord by A, B, C, X, Y , Z but B is a Borel function,

• finitely additive measures by Ξ,

• sequences of natural numbers or ordinals by η, ν, %,

s is used for various things, T is as in Definition 2.9, t is a member of T . We denote

• forcing notions by P , Q,

• forcing conditions by p, q

and use r to denote members of Random (see below) except in Definition 2.2.

Leb is the Lebesgue measure (on ^ω 2), and Random will be the family {r ⊆ ^ω> 2 : r is a subtree of ( ^ω> 2, C) (i.e. non-empty subset of ^ω> 2

closed under initial segments) with no C-maximal element (so lim(r) := {η ∈ ^ω 2 : (∀n ∈ ω)(η n ∈ t)} is a closed subset of ^ω 2) and Leb(lim(r)) > 0}

ordered by inverse inclusion. We may sometimes use instead

{B : B is a Borel non-null subset of ^ω 2}.

For η ∈ ^ω> 2 and A ⊆ ^ω≥ 2 let

A ^[η] = {ν ∈ A : ν E η ∨ η E ν}.

Let H(χ) denote the family of sets with transitive closure of cardinality

< χ, and let < ^∗ _χ denote a well ordering of H(χ).

We thank Tomek Bartoszy´ nski and Mariusz Rabus for reading and com- menting and correcting.

1. Preliminaries. We review various facts on finitely additive measures.

Definition 1.1. (1) M is the set of functions Ξ from some Boolean subalgebra P of P(ω) including the finite sets to [0, 1] _R such that:

• Ξ(∅) = 0, Ξ(ω) = 1,

• if Y, Z ∈ P are disjoint, then Ξ(Y ∪ Z) = Ξ(Y ) + Ξ(Z),

• Ξ({n}) = 0 for n ∈ ω.

(2) M ^full is the set of Ξ ∈ M with domain P(ω) and members are called finitely additive measures (on ω).

(3) We say A has Ξ-measure a (or > a, or whatever) if A ∈ dom(Ξ) and Ξ(A) is a (or > a, or whatever).

Proposition 1.2. Let a α , b α (α < α ^∗ ) be reals, 0 ≤ a α ≤ b _α ≤ 1, and let A α ⊆ ω (α < α ^∗ ) be given. The following conditions are equivalent :

(A) There exists Ξ ∈ M which satisfies Ξ(A α ) ∈ [a α , b α ] for α < α ^∗ . (B) For every ε > 0, m < ω and n < ω, and α 0 < α 1 < . . . < α n−1 < α ^∗ we can find a finite, non-empty u ⊆ [m, ω) such that for l < n,

a α

− ε ≤ |A _α

∩ u|/|u| ≤ b _α

+ ε.

(C) For every real ε > 0, n < ω and α 0 < α 1 < . . . < α n−1 < α ^∗ there are c l ∈ [a _α

− ε, b _α

+ ε] such that in the vector space R ⁿ , hc 0 , . . . , c n−1 i is in the convex hull of {% ∈ ⁿ {0, 1} : for infinitely many m ∈ ω we have (∀l < n)[%(l) = 1 ⇔ m ∈ A α

]}.

(D) Like part (A) with Ξ ∈ M ^full .

(E) Like part (B) demanding u ⊆ ω, |u| ≥ m.

P r o o f. Straightforward (in fact, in clause (E) we can omit “u ⊆ [m, ω)”).

On (C) see 2.17.

Proposition 1.3. (1) Assume that Ξ ⁰ ∈ M and for α < α ^∗ , A α ⊆ ω and 0 ≤ a α ≤ b _α ≤ 1, a _α , b α reals. The following are equivalent :

(A) There is Ξ ∈ M ^full extending Ξ 0 such that α < α ^∗ ⇒ Ξ(A α ) ∈ [a α , b α ].

(B) For every partition hB 0 , . . . , B m−1 i of ω with B _i ∈ dom(Ξ ₀ ) and

ε > 0, n < ω and α 0 < . . . < α n−1 < α ^∗ we can find a finite set u ⊆ ω such

that Ξ(B i )−ε ≤ |u∩B i |/|u| ≤ Ξ(B _i )+ε and a α

−ε ≤ |u∩A _α

|/|u| ≤ b _α

+ε.

(C) For every partition hB 0 , . . . , B m−1 i of ω with B _i ∈ dom(Ξ ₀ ) and ε > 0, n < ω and α 0 < . . . < α n−1 < α ^∗ we can find c l,k ∈ [0, 1] _R for l < n, k < m such that :

(a) P

k<m c l,k ∈ (a _α

− ε, b _α

+ ε),

(b) for each k < m and s < ω we can find u ⊆ B k with ≥ s members such that

l < n ⇒ c l,k − ε < (|u ∩ A _l ∩ B _k |/|u|)Ξ(B _k ) < c l,k + ε.

(D) for every partition hB 0 , . . . , B m−1 i of ω with B _i ∈ dom(Ξ ₀ ), ε > 0, n < ω, and α 0 , . . . , α n−1 < α ^∗ we can find c l,k ∈ [0, 1] _R for l < n, k < m such that :

(a) P

k<m c l,k ∈ [a _α

− ε, b _α

+ ε], (b) hc l,k : l < ni is in the convex hull of

{% ∈ ⁿ {0, 1} : for infinitely many i ∈ B k , we have (∀l < n)[%(l) = 1 ⇔ i ∈ A α

]}.

(2) The following are sufficient conditions for (A)–(D) above:

(E) For every ε > 0, A ^∗ ∈ dom(Ξ ₀ ) such that Ξ 0 (A ^∗ ) > 0, n < ω, α 0 < . . . < α n−1 < α ^∗ , we can find a finite, non-empty u ⊆ A ^∗ such that a α

− ε ≤ |A _α

∩ u|/|u| ≤ b _α

+ ε for l < n.

(F) For every ε > 0, n < ω, α 0 < α 1 < . . . < α n−1 < α ^∗ and A ^∗ ∈ dom(Ξ 0 ) such that Ξ 0 (A ^∗ ) > 0, the set Q

l<n [a α

− ε, b _α

+ ε] ⊆ R ⁿ is not disjoint from the convex hull of

{% ∈ ⁿ {0, 1} : for infinitely many m ∈ A ^∗ we have (∀l < n)[%(l) = 1 ⇔ m ∈ A α

]}.

(3) If in addition b α = 1 for α < α ^∗ then a sufficient condition for (A)–(E) above is

(G) if A ^∗ ∈ dom(Ξ 0 ) and Ξ(A ^∗ ) > 0 and n < ω and α 0 < . . . < α n−1 <

α ^∗ then A ^∗ ∩ T

l<n A α

6= ∅.

P r o o f. Straightforward.

Definition 1.4. (1) For Ξ ∈ M ^full and a sequence ¯ a = ha l : l < ωi of reals in [0, 1] _R (or just sup _l<ω |a _l | < ∞), define Av _Ξ (¯ a) to be

sup n X

k<k

Ξ(A k ) inf{a l : l ∈ A k } : hA _k : k < k ^∗ i is a partition of ω o

= inf n X

k<k

Ξ(A k ) sup{a l : l ∈ A k } : hA _k : k < k ^∗ i is a partition of ω o

.

(Easily proved that they are equal.)

(2) For Ξ ∈ M and A ⊆ ω such that Ξ(A) > 0 we define Ξ A (B) = Ξ(A ∩ B)/Ξ(A).

Clearly Ξ A ∈ M with the same domain, Ξ _A (A) = 1. If B ⊆ ω and Ξ(B) > 0 then we let

Av _ΞB (ha k : k ∈ Bi) = Av Ξ (ha ⁰ _k : k < ωi)/Ξ(B) where

a ⁰ _k =  a k if k ∈ B, 0 if k 6∈ B.

Proposition 1.5. Assume Ξ ∈ M ^full and a ⁱ _l ∈ [0, 1] _R for i < i ^∗ < ω, l < ω, B ⊆ ω, Ξ(B) > 0 and Av Ξ

(ha ⁱ _l : l < ωi) = b i for i < i ^∗ , m ^∗ < ω and lastly ε > 0. Then for some finite u ⊆ B \ m ^∗ we have: if i < i ^∗ , then b i − ε < P{a ⁱ _l : l ∈ u}/|u| < b i + ε.

P r o o f. Let B = S

j<j

B j be a partition of B with j ^∗ < ω such that for every i < i ^∗ we have

X

j<j

sup{a ⁱ _l : l ∈ B j }Ξ(B _j ) − X

j<j

inf{a ⁱ _l : l ∈ B j }Ξ(B _j ) < ε/2.

Now choose k ^∗ large enough such that there are k j satisfying k ^∗ = P

j<j

k j

and |k j /k ^∗ − Ξ(B _j )/Ξ(B)| < ε/(2j ^∗ ) for j < j ^∗ . Let u j ⊆ B _j \ m ^∗ with

|u _j | = k _j for j < j ^∗ , and let u = S

j<j

u j . Now calculate:

X

l∈u

a ⁱ _l /|u| = X

j<j

X {a ⁱ _l : l ∈ u j }/|u| ≤ X

j<j

sup{a ⁱ _l : l ∈ B j }k _j /k ^∗

≤ X

j<j

sup{a ⁱ _l : l ∈ B j }(Ξ(B _j )/Ξ(B) + ε/(2j ^∗ ))

≤ b _i + ε/2 + ε/2 = b i + ε, X

l∈u

a ⁱ _l /|u| = X

j<j

X {a ⁱ _l : l ∈ u j }/|u| ≥ X

j<j

inf{a ⁱ _l : l ∈ B j }k _j /k ^∗

≥ X

j<j

inf{a ⁱ _l : l ∈ B j }(Ξ(B j )/Ξ(B) − ε/(2j ^∗ )) > b i − ε.

Claim 1.6. Suppose Q ¹ , Q 2 are forcing notions, Ξ 0 ∈ M ^full in V , and for l = 1, 2,

Q

“ e

Ξ l is a finitely additive measure extending Ξ 0 ”.

Then

Q

×Q

“there is a finitely additive measure extending e Ξ 1 and

e Ξ 2

(hence Ξ 0 )”.

P r o o f. Straightforward by 1.2 as (∗) if Q

“

e

A l ⊆ ω” and ^Q

×Q

“ e A 1 ∩

e

A 2 is finite” then ^Q

×Q

“for some m and A ⊆ ω, A ∈ V we have

e

A 1 \ m ⊆ A, ( e

A 2 \ m) ∩ A = ∅”.

Fact 1.7. Assume Ξ is a partial finitely additive measure, and ¯ a ^α = ha ^α _k : k < ωi a sequence of reals for α < α ^∗ such that lim sup _k |a ^α _k | < ∞ for each α. Then (B)⇒(A) where

(A) there is Ξ ^∗ with Ξ ⊆ Ξ ^∗ ∈ M ^full such that Av Ξ

(¯ a ^α ) ≥ b α for α < α ^∗ ,

(B) for every partition hB 0 , . . . , B m

−1 i of ω with B m ∈ dom(Ξ) and ε > 0, k ^∗ > 0 and α 0 < . . . < α n

−1 < α ^∗ , there is a finite u ⊆ ω \ k ^∗ such that :

(i) Ξ(B m ) − ε < |u ∩ B m |/|u| < Ξ(B _m ) + ε, (ii) |u| ⁻¹ P

k∈u a ^α _k

> b α

− ε for l < n ^∗ .

Remark 1.8. If in (A) we demand Av Ξ (¯ a ^α ) = b α , then in (B)(ii) add

|u| ⁻¹ P

k∈u a ^α _k

≤ b α

+ ε.

2. The iteration. Ignoring MA <κ (which anyhow was a side issue) a quite natural approach in order to get 0.1 (i.e. cov(null) = λ, say λ = ℵ ω ) is to use finite support iteration, ¯ Q = hP α , Q α : α < α ^∗ i, add in the first λ steps null sets N α (the intention is that S

α<λ N α = ^ω 2 in the final model), and then iterate with Q α being Random ^V

where P _α ⁰ < ^◦ P α and |P _α ⁰ | < λ.

Say, for some A α ⊆ α,

P _α ⁰ = {p ∈ P α : dom(p) ⊆ A α and this holds for the conditions involved in the P γ -name p(γ) for γ ∈ dom(p) etc.}

(so each Q α is a partial random; see Definition 2.2). If every set of < λ null sets from V ^P

is included in some V ^P

, clearly V ^P

|= cov(null) ≥ λ; but we need the other inequality too. So we are using “non-transitive” memory, i.e. α ∈ A β 6⇒ A α ⊆ A β ; this makes our life hard.

The problem is: why does hN α : α < λi continue to cover? For P λ+n such that α ∈ [λ, λ + n) ⇒ A α = α this is very clear (we get iteration of Random forcing) and if α ∈ [λ, λ + n) ⇒ A α ⊆ λ this is clear (we get product). But necessarily we get a quite chaotic sequence hA α

∩ {α _l : l < m} : m < m ^∗ i for some α 0 < . . . < α m

−1 . More concretely this is the problem of why there are no perfect sets of random reals (see 2.7) or even just no dominating reals. We need to “let the partial randoms whisper secrets to one another”, in other words to pass information in some way. This is done by the finitely additive measures

e

Ξ ^t _α . We had tried with thinking of using ℵ ε -support (see

[Sh 538]), the idea is still clear in the proof of 3.3. In this proof we start

with “no dominating reals” for which we can just use ultrafilters (rather than finitely additive measures).

Let us start with a ground model V satisfying the following hypothesis:

Hypothesis 2.1. (a) λ = P

ζ<δ(∗) λ ζ , ℵ 0 < κ = cf(κ), κ < λ ζ < λ γ for ζ < γ < δ(∗), 2 ^κ = χ and ζ < δ(∗) ⇒ (λ ζ ) ^ℵ

< λ.

(b) We have one of the following ( ¹ ):

(α) cf(χ) > λ, the length of the final iteration is χ,

(β) the length of the final iteration is χ×χ×λ ⁺ and cf([χ] ^<λ , ⊆) = χ.

We speak mainly on (α). In case (β) we should be careful to have no repetitions in ¯ η = hη α : α < δ ^∗ i (see below) or hη _α /≈ κ : α < δ ^∗ i with no repetitions, where η ≈ κ ν iff η, ν ∈ ^κ 2 and |{i < κ : η(i) 6= ν(i)}| < κ.

The reader may choose to restrict himself and start with V satisfying:

GCH, λ = ℵ ω , δ(∗) = ω, λ n = ℵ _n(∗)+n , κ = ℵ _n(∗) > ℵ 1 and χ = ℵ ω+1 . Now add ℵ ω+1 generic subsets of κ, i.e., force with a product of χ copies of ( ^κ> 2, C) with support < κ. This model satisfies the hypothesis.

We intend to define a forcing P such that

V ^P |= 2 ^ℵ

= χ + cov(null) = λ + MA <κ . Definition 2.2. (1) K is the family of sequences

Q = (P ¯ α , e

Q α , A α , µ α ,

˜ τ α : α < α ^∗ ) satisfying:

(A) (P α , e

Q α : α < α ^∗ ) is a finite support iteration of c.c.c. forcing no- tions, we set α ^∗ = lg( ¯ Q) (the length of ¯ Q), P α

is the limit.

(B) τ ˜ α ⊆ µ _α < κ is the generic of e

Q α (i.e. over V ^P

from G e

Q

we can compute

˜ τ α and vice versa).

(C) A α ⊆ α (for proving Theorem 0.1 we use |A α | < λ).

(D) e

Q α is a P α -name of a c.c.c. forcing notion but computable from h ˜ τ γ [

e

G P

] : γ ∈ A α i; in particular it belongs to V _α = V[h

˜ τ γ [G P

] : γ ∈ A α i].

(E) α ^∗ ≥ λ and for α < λ we have Q α = ( ^ω> 2, C) (the Cohen forcing) and µ α = ℵ 0 (well, identifies ^ω> 2 with ω).

(F) For each α < α ^∗ one of the following holds (and the case is deter- mined in V, not just a P α -name):

(α) | e

Q α | < κ, |A α | < κ and (just for notational simplicity) the set of elements of

e

Q α is µ α < κ (but the order not necessarily the order of the ordinals) and

e

Q α is separative (i.e. ζ ξ ∈ G e

Q

⇔ e

Q α |= ξ ≤ ζ).

In this case let

˜ τ α = e G

e

Q

.

( ¹ ) Actually, any ordinal α ^∗ of cardinality χ, divisible by χ and of cofinality > λ is

O.K.

(β) Essentially e

Q α = Random ^V

(= {r ∈ V α : r ⊆ ^ω> 2, perfect tree, Leb(lim(r)) > 0}) and |A α | ≥ κ; but for simplicity

e Q α = Random ^A

^{, ¯ Qα} where for A ⊆ lg( ¯ Q),

Random ^{A, ¯ Q} = {p : there is (in V) a Borel function B = B(x 0 , x 1 , . . .), with variables ranging on {true, false} and range per- fect subtrees r of ^ω> 2 with Leb(lim(r)) > 0 such that (∀η ∈ r)[Leb(lim(r ^[η] )) > 0] (recall r ^[η] = {ν ∈ r : ν E η ∨ η E ν}), and there are pairs (γ l , ζ l ) for l < ω, with γ l ∈ A and ζ _l < µ γ

, such that p = B(. . . , truth value(ζ l ∈

˜ τ γ

), . . .) l<ω } (in other notation, p = B(truth value(ζ l ∈

˜ τ γ

) : l < ω)) and then we let supp(p) = {γ l : l < ω}. In this case µ α = ω and

˜ τ α is the random real, i.e.

τ ˜ α (n) = l ⇔ (∃η ∈ ⁿ 2)[( ^ω> 2) ^[η

^hli] ∈ e G Q

].

(2) Let

P _α ⁰ = {p ∈ P α : for every γ ∈ dom(p), if |A γ | < κ then p(γ) is an ordinal

< µ γ (not just a P γ -name) and if |A γ | ≥ κ then p(γ) has the form mentioned in clause (F)(β) above (and not just a P γ -name of such an object)}

(this is a dense subset of P α ).

(3) For A ⊆ α let

P _A ⁰ = {p ∈ P _α ⁰ : dom(p) ⊆ A and γ ∈ dom(p) ⇒ supp(p(γ)) ⊆ A}.

Fact 2.3. Suppose ¯ Q ∈ K with lg( ¯ Q) = α ^∗ .

(1) For α ≤ α ^∗ , P _α ⁰ is a dense subset of P α and P α satisfies the c.c.c.

(2) Suppose

(a) cf(α ^∗ ) > λ,

(b) for every A ⊆ α ^∗ , if |A| < λ, then there is β < α ^∗ such that A ⊆ A β (and |A β | ≥ κ).

Then, in the extension, ^ω 2 is not the union of < λ null sets.

(3) In V ^P

, from

τ ˜ α [G Q

] we can reconstruct G Q

and vice versa. From h ˜ τ γ : γ < αi[G P

] we can reconstruct G P

and vice versa. So V ^P

= V[h

τ ˜ β : β < αi].

(4) If µ < λ, and e

X is a P α

-name of a subset of µ, then there is a set A ⊆ α ^∗ such that |A| ≤ µ and P

“

e

X ∈ V[h

τ ˜ γ : γ ∈ Ai]”. Moreover for each ζ < µ there is in V a Borel function B(x 0 , . . . , x n , . . .) n<ω with domain and range the set {true, false} and γ l ∈ A, ζ l < µ γ

for l < ω such that

^P

“ζ ∈ e

X iff true = B(. . . , “truth value of ζ l ∈

˜ τ γ

[G Q

]”, . . .) l<ω ”.

(5) For ¯ Q ∈ K and A ⊆ α ^∗ , every real in V[h

˜ τ γ : γ ∈ Ai] has the form B(. . . , truth value(ζ _l ∈ τ _γ

), . . .) l<ω where γ l < α ^∗ , ζ l < µ γ

and B is a Borel function (from V) from ^ω {true, false} to ^ω 2 (the set of reals).

(6) If condition (c) below holds then V ^P

|= MA _<κ , where (c) if

e

Q is a P α

-name of a c.c.c. forcing notion with set of elements µ < κ then for some α < α ^∗ ,

e

Q is a P α -name µ α = µ and P

“

e Q =

e Q α ”.

(7) If |A β | ≥ κ and P _A ⁰

β

< ^◦ P β then Q β is actually Random ^V

. (8) If |A α | < λ then P

“

e

Q α < λ”, in fact |{p(α) : p ∈ P _α ⁰

}| < λ.

P r o o f. (1) Easy, by induction on α.

(2) Easy using parts (3)–(7). Note that for any β < α ^∗ satisfying |A β | ≥ κ the null sets from V ^[h ˜ ^τ

^:γ∈A

^i] do not cover ^ω 2 in V ^P

as we have random reals over V ^{[h τ} ˜

^:γ∈A

^i] . So, by clause (b) of the assumption, it is enough to note that if

e

y is a P α

-name of a member of ^ω 2, then there is a countable A ⊆ α ^∗ such that

e y[

e

G] ∈ V[h

τ ˜ β : β ∈ Ai]. This follows by part (4).

(3) By induction on α.

(4) Let χ ^∗ be such that { ¯ Q, λ} ∈ H(χ ^∗ ), and let ζ < µ; let M be a count- able elementary submodel of (H(χ ^∗ ), ∈, < ^∗ _χ

) to which { ¯ Q, λ, κ, µ,

e

X, ζ} be- longs, so ^P

“M [

e

G P

] ∩ H(χ ^∗ ) = M ”. Hence by 2.3(3) (i.e. as V ^P

= V[h ˜ τ β : β < αi]) we have M [

e

G P

] = M [h

τ ˜ i : i ∈ α ^∗ ∩M i] and the conclusion should be clear.

(5) By 2.3(4).

(6) Straight.

(7) Easy (see more [Sh 619, §3]).

(8) By the definitions (and since (λ ζ ) ^ℵ

< λ and λ = sup{λ ζ : ζ < δ(∗)}

by 2.1).

Definition 2.4. (1) Suppose that ¯ a = ha l : l < ωi and hn l : l < ωi are such that:

(a) a l ⊆ ⁿ

2,

(b) n l < n l+1 < ω for l < ω, (c) |a l |/2 ⁿ

> 1 − 1/10 ^l .

Let N [¯ a] =: {η ∈ ^ω 2 : (∃ ^∞ l)(∀ν ∈ a l )(ν 6 C η)}.

(2) For ¯ a as above and n ∈ ω, let

tree n (¯ a) = {ν ∈ ^ω> 2 : n l > n ⇒ ν n ^l ∈ a l }.

It is well known that for ¯ a as above the set N [¯ a] is null (and N [¯ a] =

ω 2 \ S

n<ω lim(tree n (¯ a))).

Definition 2.5. For α < λ we identify Q α (the Cohen forcing) with

{h(n _l , a l ) : l < ki : k < ω, n l < n l+1 < ω, a l ⊆ ⁿ

2, |a l |/2 ⁿ

> 1 − 1/10 ^l },

ordered by end extension. If G Q

is Q α -generic, let h(¯

e a ^α _l ,

e

n ^α _l ) : l < ωi[G Q

] be the ω-sequence such that every p ∈ G Q

is an initial segment of it. So we have defined the Q α -name ¯

e a ^α = h

e

a ^α _l : l < ωi and similarly h e

n ^α _l : l < ωi.

Let e

N α = N [¯

e a ^α ].

Our aim is to prove (∗) Q ¯ , where

Definition 2.6. For ¯ Q ∈ K with α ^∗ = lg( ¯ Q) let

(∗) Q ¯ hN _α : α < λi cover ^ω 2 in V ^P

, where P α

= Lim( ¯ Q).

We shall eventually prove it not for every ¯ Q, but for enough ¯ Q’s (basically asking the A α of cardinality ≥ κ to be closed enough).

Lemma 2.7. For ¯ Q ∈ K with γ = lg( ¯ Q), a sufficient condition for (∗) Q ¯ is:

(∗∗) Q ¯ In V ^P

, there is no perfect tree T ⊆ ^ω> 2 and E ∈ [λ] ^κ

such that , for some n < ω, T ⊆ tree n [¯ a ^α ] for all α ∈ E.

P r o o f. By induction on γ ≥ λ. For γ = λ, trivial by properties of the Cohen forcing.

Suppose γ > λ is limit. Assume toward contradiction that p P

“

e η 6∈ [

α<λ

N [¯

e a ^α ]”.

Without loss of generality,

p P

“ e

η 6∈ V ^P

”

for every β < γ, hence by properties of FS iteration of c.c.c. forcing notions, cf(γ) = ℵ 0 . So for each α < λ there are p α , m α such that

p ≤ p α ∈ P _γ , p α “ e

η ∈ lim(tree m

(¯

e a ^α ))”.

Note that (by properties of c.c.c. forcing notions) h{α < λ : p α ∈ P _β } : β < γi is an increasing sequence of subsets of λ of length γ, so for some γ 1 < γ there is E ∈ [λ] ^κ

such that p α ∈ P _γ

for every α ∈ E and we can assume m α = m for α ∈ E. Note that for all but < κ ⁺ of the ordinals α ∈ E we have

p α “|{β ∈ E : p β ∈ G _P

γ1

}| = κ ⁺ ”.

Fix such an α, and let G P

be a P γ

-generic (over V) subset of P γ

to which p α belongs. Now in V[G P

] let E ⁰ = {β ∈ E : p β ∈ G P

} so |E ⁰ | = κ ⁺ . Let T ^∗ = T

β∈E

tree m (¯ a ^β ). Note that, in V ^P

, T ^∗ is a subtree of ^ω> 2 and by (∗∗) Q ¯ , T ^∗ contains no perfect subtree. Hence lim(T ^∗ ) is countable, so absolute. But p α P

“

e η ∈ lim(

e

T ^∗ )”, so p α “ e

η ∈ V ^P

”, a contradiction.

Assume now that γ = β + 1 > λ and work in V ^P

. Choose p, p α ∈ Q _β

as before. By 2.3(7), Q β has a dense subset of cardinality < λ, so there are

q ∈ Q β and m such that E = {α < λ : m α = n, p α ≤ q} has cardinality

≥ κ ⁺ . Continue as above.

As we have covered the cases γ = λ, γ > λ limit and γ > λ successor, we have finished the proof.

Discussion 2.8. Note that by Lemma 2.7 and Fact 2.3 it is enough to show that there is ¯ Q ∈ K such that α ^∗ = lg( ¯ Q) (where α ^∗ is chosen as the length of the final iteration from 2.1(b)), satisfying clauses (a)+(b)+(c) of Fact 2.3(2)+(6) and (∗∗) Q ¯ . To prove the latter we need to impose more restrictions on the iteration. Now when Haim Judah asked me on the prob- lem, whereas 2.1–2.7 were quite immediate, arriving at 2.9–2.11 has taken me many years and much effort.

Definition 2.9. T , the set of blueprints, is the set of tuples t = (w ^t , n ^t , m ^t , ¯ η ^t , h ^t ₀ , h ^t ₁ , h ^t ₂ , ¯ n ^t )

where:

(a) w ^t ∈ [κ] ^ℵ

,

(b) 0 < n ^t < ω, m ^t ≤ n ^t ,

(c) ¯ η ^t = hη _n,k ^t : n < n ^t , k < ωi, η _n,k ^t ∈ ^w

2 for n < n ^t , k < ω,

(d) h ^t ₀ is a partial function from [0, n ^t ) to ^ω κ, its domain includes the set {0, . . . , m ^t − 1} (here we consider members of Q _α for α < λ as integers ( ² )), (e) h ^t ₁ is a partial function from [0, n ^t ) to (0, 1) _Q (rationals), but for n ∈ [0, n ^t ) \ dom(h ^t ₁ ) we stipulate h ^t ₁ (n) = 0 and we assume P

n<n

ph ^t ₁ (n) <

1/10.

(f) h ^t ₂ is a partial function from [0, n ^t ) to ^ω> 2, (g) dom(h ^t ₀ ), dom(h ^t ₁ ) are disjoint with union [0, n ^t ), (h) dom(h ^t ₂ ) = dom(h ^t ₁ ),

(i) η ^t _n

_,k

= η _n ^t

_,k

⇒ n ₁ = n 2 ,

(j) for each n < n ^t we have: hη _n,k ^t : k < ωi is constant or with no repetitions; if it is constant and n ∈ dom(h ^t ₀ ) then h ^t ₀ (n) is constant,

(k) ¯ n ^t = hn ^t _k : k < ωi where n ^t ₀ = 0, n ^t _k < n ^t _k+1 < ω and the sequence hn ^t _k+1 − n ^t _k : k < ωi goes to infinity. For l < ω and such ¯ n let k n ¯ (l) = k(l, ¯ n) be the unique k such that n k ≤ l < n k+1 .

Discussion 2.10. The definitions of a blueprint t ∈ T (in Definition 2.9) and of iterations ¯ Q ∈ K ³ (defined in Definition 2.11(c) below; the reader may first read it) contain the main idea of the proof, so though they have many clauses, the reader is advised to try to understand them.

( ² ) Actually the case where each h ^t ₀ (n) is a constant function from ω to κ suffices,

and so κ < λ suffices instead of κ ^ℵ

< λ.

In order to prove (∗∗) Q ¯ we will show in V ^P

that if E ∈ [λ] ^κ

and n < ω, then T

α∈E tree n (¯

e

a ^α ) is a tree with finitely many branches. So let p be given, let p ≤ p ζ “β ^ζ ∈

e

E” for ζ < κ ⁺ , β ζ 6∈ {β ξ : ξ < ζ}, we can assume p ζ is in some pregiven dense set, and hp ζ : ζ < κ ⁺ i form a ∆-system (with some more “thinning” demands), dom(p ζ ) = {α n,ζ : n < n ^∗ }, α _n,ζ is increasing with n, and α n,ζ < λ iff n < m ^∗ . Let p ⁰ _ζ be p ζ when p ζ (α n,ζ ) is increased a little, as described below.

It suffices to find p ^∗ ≥ p such that p ^∗ “ e

A := {ζ < ω : p ⁰ _ζ ∈ e

G} is large enough such that T

ζ∈

e

A tree m (¯ a ^β

) has only finitely many branches”.

Because of “communication problems” the “large enough” is interpreted as of

e

Ξ ^t _α -measure (again defined in 2.11 below).

The natural numbers n < n ^∗ such that Q α

is a forcing notion of cardinality < κ do not cause problems, as h ^t ₀ (n) tells us exactly what the condition p ζ (α n,ζ ) is. Still there are many cases of such hp ζ : ζ < ωi which fall into the same t; we possibly will get contradictory demands if α n

,ζ

= α n

,ζ

, n 1 6= n ₂ . But the w ^t , ¯ η ^t are exactly built to make this case not to happen. That is, we have to assume 2 ^κ = χ (= |α ^∗ |) in order to be able for our iteration hP α ,

e

Q α : α < α ^∗ i to choose hη _α : α < α ^∗ i, η _α ∈ ^κ 2, with no repetitions, so that if v ⊆ χ, |v| ≤ ℵ 0 (e.g. v = {α n,ζ : n < n ^t , ζ < ω}) then for some w = w ^t ∈ [κ] ^ℵ

we have hη α w : α ∈ vi with no repetitions.

So the blueprint t describes such a situation, giving as much information as possible, as long as the number of blueprints is not too large, κ ^ℵ

= κ in our case.

If Q α

is a partial random, we may get many candidates for p ζ (α n,ζ ) ∈ Random and they are not all the same ones. We want them in many cases to be in the generic set. Well, we can (using h ^t ₁ (n), h ^t ₂ (n)) know that in some interval ( ^ω 2) ^[h

^(n)] the set lim(p ζ (α n,ζ )) is large, say of relative measure

≥ 1 − h ^t ₁ (n), and we could have chosen the p ζ ’s such that hh ^t ₁ (n) : n < n ^t i is small enough, still the number of candidates is not bounded by 1/h ^t ₁ (n).

Here taking limit along ultrafilters is not good enough, but using finitely additive measures is.

Well, we have explained w ^t , ¯ η ^t , h ^t ₀ , h ^t ₁ , h ^t ₂ , but what about the ¯ n ^t = hn ^t _k : k < ωi? In the end (in §3) the specific demand on {ζ : p ⁰ _ζ ∈ G} being large, is that for infinitely many k < ω,

|{l : n ^t _k ≤ l < n ^t _k+1 and p l ∈ G}|/(n ^t _k+1 − n ^t _k )

is large, the n ^t _k will be chosen such that it is increasing fast enough and hp ⁰ _l (β l ) : l ∈ [n ^t _k , n ^t _k+1 )i will be chosen such that for each ε > 0 for some s < ω, for k large enough if the above fraction is ≥ ε then essentially

k 2 ∩ {tree m (¯ a ^β

) : n ^t _k ≤ l < n ^t _k+1 and p ⁰ _l ∈ G} has ≤ s members; this

suffices.

What is our plan? We define K ³ , the class of suitable expanded iterations Q by choice of η ¯ α (for α < lg( ¯ Q)) and names for finitely additive measures

e

Ξ ^t _α satisfying the demands natural in this context. You may wonder why we use Ξ-averages; this is like integral or expected value, and so they “behave nicely” making the “probability computations” simpler. Then we show that we can find ¯ Q ∈ K ³ in which all obligations toward “cov(null) ≥ λ” and MA <κ hold.

The main point of §3 will be that we can carry out the argument of

“for some p ^∗ we have p ^∗ {l < ω : p ⁰ l ∈ G} is large” and why it gives n < ω & E ∈ [χ] ^κ

⇒ T

ζ∈A tree n (¯ a ^ζ ) has finitely many branches, thus proving Theorem 0.1.

The reader may wonder how much the e

Ξ ^t _α are actually needed. As explained above they are just a transparent way to express the property;

this will be utilized in [Sh 619].

Definition 2.11. K ³ is the class of sequences Q = hP ¯ α ,

e

Q β , A β , µ β ,

τ ˜ β , η β , ( e

Ξ ^t _α ) t∈T : α ≤ α ^∗ , β < α ^∗ i (we write α ^∗ = lg( ¯ Q)) such that:

(a) hP α , e

Q β , A β , µ β ,

˜ τ β : α ≤ α ^∗ , β < α ^∗ i is in K, (b) η β ∈ ^κ 2 and (∀β < α < α ^∗ )[η α 6= η _β ],

(c) T is from Definition 2.9, and e

Ξ ^t _α is a P α -name of a finitely additive measure on ω (in V ^P

, i.e. ∈ (M ^full ) ^V

), increasing with α,

(d) we say that ¯ α = hα l : l < ωi satisfies (t, n) (for ¯ Q) if:

• hα _l : l < ωi ∈ V (of course),

• t ∈ T , n < n ^t ,

• α _l ≤ α _l+1 < α ^∗ ,

• n < m ^t ⇔ (∀l)(α _l < λ) ⇔ (∃k)(α k < λ),

• η ^t _n,l ⊆ η _α

(as functions),

• if n ∈ dom(h ^t ₀ ), then µ α

< κ and P

“|Q α

| < κ and (h ^t ₀ (n))(l) ∈

e

Q α

, i.e. (h ^t ₀ (n))(l) < µ α

”,

• if n ∈ dom(h ^t ₁ ), then µ α

≥ κ so P

“Q α

has cardinality ≥ κ”

(so it is a partial random),

• if hη _n,k ^t : k ∈ ωi is constant, then (∀l)[α l = α 0 ]; if hη _n,k ^t : k < ωi is not constant, then (∀l)[α l < α l+1 ],

(e) if ¯ α = hα l : l < ωi satisfies (t, n) for ¯ Q, V

l<ω (α l < α l+1 ), n ∈ dom(h ^t ₀ ) then

P

“the following set has e

Ξ ^t _α

-measure 1:

{k < ω : if l ∈ [n ^t _k , n ^t _k+1 ) then (h ^t ₀ (n))(l) ∈

e

G Q

}”,

(f) if ¯ α = hα l : l < ωi satisfies (t, n) for ¯ Q, n ∈ dom(h ^t ₁ ), (∀l < ω)[α l <

α l+1 ], and ¯ r = h e

r l : l < ωi where for l < ω, e

r l is a P α

-name of a member of e

Q α

such that (it is forced that)

(∗) 1 − h ^t ₁ (n) ≤ Leb({η ∈ ^ω 2 : h ^t ₂ (n) C η ∈ lim(

e

r l )})/2 ^lg(h

⁽ⁿ⁾⁾ , then for each ε > 0 we have

^P

“the following set has e

Ξ ^t _α

-measure 1:

{k < ω : in the set {l ∈ [n ^t _k , n ^t _k+1 ) : e r l ∈ G _Q

αl

} there are at least (n ^t _k+1 − n ^t _k )(1 − h ^t ₁ (n))(1 − ε) elements}”, (g) if ¯ α = hα l : l < ωi satisfies (t, n) for ¯ Q, n ∈ dom(h ^t ₁ ), (∀l)[α l = α], and

e r,

e

r l are P _A ⁰

-names of members of Q α satisfying (∗∗) ^{Q ¯}

˜ ^r,h ˜ ^r

^:l<ωi

(see below for the definition of (∗∗)) then

^P

“if e r ∈

e

G Q

then 1 − h ^t ₁ (n) ≤ Av

e

Ξ

(h|{l ∈ [n ^t _k , n ^t _k+1 ) : e r l ∈

e

G Q

}|/(n ^t _k+1 − n ^t _k ) : k < ωi)”, where

(∗∗) ^{Q ¯}

˜ ^r,h ˜ ^r

^:l<ωi e r,

e

r l are P _A ⁰

-names of members of Q α , h e

r l : l < ωi ∈ V and, in V ^P

, for every r ⁰ ∈ Q _α satisfying r ≤ r ⁰ we have

Av Ξ

(ha k (r ⁰ ) : k < ωi) ≥ 1 − h ^t ₁ (n) where

a k (r ⁰ ) = a k (r ⁰ , ¯ r) = a k (r ⁰ , ¯ r, ¯ n ^t ) ( )

=

 X

l∈[n

,n

)

Leb(lim(r ⁰ ) ∩ lim(r l )) Leb(lim(r ⁰ ))

 1

n ^t _k+1 − n ^t _k

(so a k (r ⁰ , ¯ r, ¯ n) ∈ [0, 1] _R is well defined for k < ω, ¯ r = hr l : l < ωi, {r, r l } ⊆ Random, ¯ n = hn l : l < ωi, n l < n l+1 < ω),

(h) |A α | ≥ κ ⇒ P _A ⁰

< ^◦ P α , and ( ³ ) β ∈ A α & |A β | < κ ⇒ A _β ⊆ A _α , (i) if P

“|

e

Q α | ≥ κ”, then e

Ξ ^t _α P(ω) ^V

is a P A

-name ( ⁴ ) for every t ∈ T .

Definition 2.12. (1) For ¯ Q ∈ K ³ and for α ^∗ ≤ lg( ¯ Q) let Q ¯ α ^∗ = hP α ,

e

Q β , A β , µ β ,

˜ τ β , η β , (Ξ _α ^t ) t∈T : α ≤ α ^∗ , β < α ^∗ i.

(2) For ¯ Q ¹ , ¯ Q ² ∈ K ³ we say: ¯ Q ¹ ≤ ¯ Q ² if ¯ Q ¹ = ¯ Q ²  lg( ¯ Q ¹ ).

Fact 2.13. (1) If ¯ Q ∈ K ³ and α ≤ lg( ¯ Q), then ¯ Q α ∈ K ³ . (2) (K ³ , ≤) is a partial order.

( ³ ) If we add “(∀α < κ)(|α| ^ℵ

< κ)” to 2.1, we can simplify and assume α < α ^∗ ⇒ P _A ⁰

α

< ◦ P α .

( ⁴ ) Here the secret was whispered.

(3) If a sequence h ¯ Q ^β : β < δi ⊆ K ³ is increasing and cf(δ) > ℵ 0 , then there is a unique ¯ Q ∈ K ³ which is the least upper bound , lg( ¯ Q) = S

β<δ lg( ¯ Q ^β ) and ¯ Q ^β ≤ ¯ Q for all β < δ.

P r o o f. Easy (recall that it is well known that ( ^ω 2) ^V

= S

β<δ ( ^ω 2) ^V

, so

e Ξ ^t _δ = S

β<δ Ξ _β ^t is a legal choice; this is the use of cf(δ) > ℵ 0 ).

Lemma 2.14. Suppose that ¯ Q ⁿ ∈ K ³ , ¯ Q ⁿ < ¯ Q ⁿ⁺¹ , α n = lg( ¯ Q ⁿ ), δ = sup _n<ω α n . Then there is ¯ Q ∈ K ³ such that lg( ¯ Q) = δ and ¯ Q ⁿ ≤ ¯ Q for n ∈ ω.

P r o o f. Note that the only problem is to define e

Ξ ^t _δ for t ∈ T , i.e., we have to extend S

α<δ

e

Ξ ^t _α so that the following two conditions are satisfied (they correspond to clauses (f) and (e) of Definition 2.11).

(a) We are given ( ⁵ ) n < n ^t , n ∈ dom(h ^t ₁ ), hα l : l < ωi (from V of course) satisfies (t, n) for ¯ Q and is strictly increasing with limit δ and we are given h

e

p l : l < ωi such that P

“ e p l ∈

e

Q α

and 1 − h ^t ₁ (n) ≤ Leb({η ∈

ω 2 : h ^t ₂ (n) C η ∈ lim(

e

p l )})/2 ^lg(h

⁽ⁿ⁾⁾ ”. The demand is: for each ε > 0 we have P

“

e Ξ ^t _δ (

e

C) = 1”, where

e

C = {k < ω : in the set {l : l ∈ [n ^t _k , n ^t _k+1 ) and p l ∈ e

G Q

} there are at least (n ^t _k+1 − n ^t _k )(1 − h ^t ₁ (n))(1 − ε) elements}.

(b) If ( ⁴ ) n < n ^t , n ∈ dom(h ^t ₀ ), hα l : l < ωi satisfies (t, n) for ¯ Q and is strictly increasing with limit δ, and p l ∈ Q α

satisfy p l = h ^t ₀ (n)(l) for l < ω (an ordinal < µ α

), then P

“

e Ξ ^t _δ (

e

C) = 1” where

e

C = {k < ω : for every l ∈ [n k , n k+1 ) we have p l ∈ e G Q

}.

As S

α<δ

e

Ξ ^t _α is a (P δ -name of a) member of M in V ^P

, by 1.3(3) it suffices to prove

(∗) P

“if e

B ∈ S

α<δ dom(

e

Ξ ^t _α ) = S

α<δ P(ω) ^V[P

^] and e Ξ ^t _α (

e B) > 0 and j ^∗ < ω, and

e

C j (for j < j ^∗ ) are from (a), (b) above then e

B ∩ T

j<j

e

C j 6= ∅”.

Toward contradiction assume q ∈ P δ force the negation, so possibly increas- ing q we have: for some

e

B and for some j ^∗ < ω, for each j < j ^∗ we have the ε j > 0, and n(j) < n ^t , hα ^j _l : l < ωi, hp ^j _l : l < ωi involved in the definition of

e

C j (in (a) or (b) above), q force:

e B ∈ S

α<δ dom(

e

Ξ ^t _α ) = S

α<δ P(ω) ^V[P

^] and ( S

α<δ

e Ξ ^t _α )(

e

B) > 0 and e

C j (for j < j ^∗ ) comes from (a) or (b) above, but e

B ∩ T

j<j

e

C j = ∅; as we can decrease ε, without loss of generality ε j = ε.

Again we can assume that for some α(∗) < δ, e

B ∈ dom(

e

Ξ ^t _α(∗) ) is a P α(∗) - name, and

e

C j have the n(j) < n ^t , hα ^j _l : l < ωi, hp ^j _l : l < ωi witnessing it is as

( ⁵ ) In V, so h(α _l , p _l ) : l < ωi ∈ V, of course.

required in (a) or (b) above. Without loss of generality, q ∈ P _α(∗) . Possibly increasing q (inside P α(∗) though) we can find k < ω such that q “k ∈

e B”

and V

j<j

V

l∈[n

,n

) α ^j _l > α(∗) and moreover such that n ^t _k+1 − n ^t _k is large enough compared to 1/ε, j ^∗ (just let q ∈ G P

⊆ P _α(∗) , G P

generic over V and think in V[G P

]). Let {α ^j _l : j < j ^∗ and l ∈ [n ^t _k , n ^t _k+1 )} be listed as {β m : m < m ^∗ }, in increasing order (so β ₀ > α(∗)) (possibly α ^j(1) _l(1) = α ^j(2) _l(2) & (j(1), l(1)) 6= (j(2), l(2))). Now we choose by induction on m ≤ m ^∗ a condition q m ∈ P _β

above q, increasing with m, where we stipulate β m

= δ.

During this definition we “throw a dice” and prove that the probability of success (i.e. q m

“k ∈

e

C j ” for j < j ^∗ ) is positive, so there is q m

as desired and hence we get the desired contradiction.

Case A: m = 0. Let q ⁰ = q.

Case B: m + 1, and for some n < n ^t , we have n ∈ dom(h ^t ₀ ) and ζ and:

if j < j ^∗ and l < ω then α ^j _l = β m ⇒ n(j) = n & p ^j _l = ζ (= h ^t ₀ (n(j))(l)) ∈ Q β

). In this case dom(q m+1 ) = dom(q m ) ∪ {β m }, and

q m+1 (β) =  q m (β) if β < β m (so β ∈ dom(q m )), ζ if β = β m .

Case C: m + 1 and for some n < n ^t , we have n ∈ dom(h ^t ₁ ) and: α ^j _l = β m ⇒ n(j) = n. Work first in V[G _P

], q m ∈ G _P

, G P

generic over V.

The sets {lim(

e

p ^j _l [G P

]) ∩ ( ^ω 2) ^[h

^(n)] : α ^j _l = β m (and l ∈ [n ^t _k , n ^t _k+1 ), j < j ^∗ )}

are subsets of ( ^ω 2) ^[h

^(n)] = {η ∈ ^ω 2 : h ^t ₂ (n) C η}, and we can define an equivalence relation E m on ( ^ω 2) ^[h

^(n)] :

ν 1 E m ν 2 iff ν 1 ∈ lim(

e

p ^j _l [G P

]) ≡ ν 2 ∈ lim(

e

p ^j _l [G P

]) whenever α ^j _l = β m .

Clearly E m has finitely many equivalence classes, call them hZ _i ^m : i < i ^∗ _m i;

all are Borel (sets of reals), hence measurable; without loss of generality, Leb(Z _i ^m ) = 0 ⇔ i ∈ [i ^⊗ _m , i ^∗ _m ), so clearly i ^⊗ _m > 0. For each i < i ^⊗ _m there is r = r m,i ∈

e

Q β

[G P

] such that

lim(p ^j _l [G P

]) ⊇ Z _i ^m ⇒ r ≥ p ^j _l [G P

],

lim(p ⁱ _l [G P

]) ∩ Z _i ^m = ∅ ⇒ lim(r) ∩ lim(p ⁱ _l [G P

]) = ∅.

We can also find a rational a m,i ∈ (0, 1) _R such that

a m,i < Leb(Z _i ^m )/2 ^{− lg(h}

⁽ⁿ⁾⁾ < a m,i + ε/(2i ^∗ _m ).

We can find q _m ⁰ ∈ G _P

, q m ≤ q _m ⁰ , such that q _m ⁰ forces all this information (so for

e Z _i ^m ,

e

r m,i we shall have P β

-names, but a m,i , i ^⊗ _m , i ^∗ _m are actual objects). We can then find rationals b m,i ∈ (a m,i , a m,i + ε/2) such that P

i<i

b m,i = 1.

Now we throw a dice choosing i m < i ^⊗ _m with the probability of i m = i being b m,i and define q m+1 as:

dom(q m+1 ) = dom(q _m ⁰ ) ∪ {β m },

q m+1 (β) =  q _m ⁰ (β) if β < β m (so β ∈ dom(q _m ⁰ )), e

r m,i

if β = β m .

An important point is that this covers all cases (and in Case B the choice of (j, l) is immaterial) as for each β m there is a unique n < n ^t and l such that η β

w ^t = η ^t _n,l (see Definitions 2.11(b) and 2.9(i)). Basic probability computations (for n ^t _k+1 − n ^t _k independent experiments) show that for each j coming from clause (a), by the law of large numbers, as k was chosen such that n ^t _k+1 − n ^t _k is large enough compared to 1/ε and j ^∗ , the probability of successes is > 1 − 1/j ^∗ , successes meaning q m

“k ∈

e

C j ” (remember that if j comes from clause (b) we always succeed).

Remark 2.15. In the definition of t ∈ T (i.e. 2.9) we can add η ^t n,ω ∈ ^w

2 (i.e. replace hη _n,l ^t : l < ωi by hη ^t _n,l : l ≤ ωi) and demand

(l) if ζ ∈ w ^t then for every n < ω large enough, η ^t _n,n (ζ) = η ^t _n,ω (ζ), and in Definition 2.11(d) use ¯ α = hα l : l < ωi but this does not help here.

Lemma 2.16. (1) Assume:

(a) ¯ Q ∈ K ³ , ¯ Q = hP α , e

Q β , A β , µ β , e r β , η β , (

e

Ξ ^t _α ) t∈T : α ≤ α ^∗ , β < α ^∗ i, (b) A ⊆ α ^∗ , κ ≤ |A| < λ,

(c) η ∈ ( ^κ 2) ^V \ {η _β : β < α ^∗ },

(d) (∀α ∈ A)[|A α | < κ ⇒ A _α ⊆ A] and P _A ⁰ < ^◦ P α

, e Q =

e

Q ^{A, ¯ Q} is the P α

-name from 2.2(F)(β) and if t ∈ T then

e

Ξ ^t _α

V ^P

is a P _A ⁰ -name.

Then there is Q ¯ ⁺ = hP α ,

e

Q β , A β , µ β ,

˜ τ β , η β , ( e

Ξ ^t _α ) t∈T : α ≤ α ^∗ + 1, β < α ^∗ + 1i from K ³ , extending ¯ Q, such that

e Q α

=

e

Q, A α

= A, η α

= η.

(2) If clauses (a)+(b)+(c) of (1) hold then we can find A ⁰ such that A ⊆ A ⁰ ⊆ α ^∗ , |A ⁰ | ≤ (|A| + κ) ^ℵ

(which is < λ by Hypothesis 2.1) and ¯ Q, A ⁰ , η satisfy (a)+(b)+(c)+(d).

P r o o f. Part (2) is easy.

(1) As before the problem is to define e

Ξ ^t _α

+1 . We have to satisfy clause

(g) of Definition 2.11 for each fixed t ∈ T . Let n ^∗ be the unique n < n ^t

such that η w ^t = η _n,l ^t . If n ^∗ ∈ dom(h ^t ₀ ) or hη _n ^t

,l : l < ωi is not constant or

there is no such n ^∗ then we have nothing to do. So assume that α l = α ^∗ and

η _n ^t

,l = η w ^t for l < ω. Let Γ (∈ V) be the set of all pairs ( e r, h

e

r l : l < ωi) which satisfy the assumption (∗∗) ^{Q ¯}

˜ ^r,h ˜ ^r

^:l<ωi

of 2.11(g). In V ^P

we have to choose

e

Ξ ^t _α

+1 taking care of all these obligations. We work in V ^P

. By assumption (d) and Claim 1.6 it suffices to prove it for V ^P

so Q α

is Random ^V

(see 2.3(7)). By 1.7 it is enough to prove condition (B) of 1.7.

Suppose it fails. Then there are hB m : m < m(∗)i, a partition of ω from V ^P

, for simplicity Ξ _α ^t

(B m ) > 0 for m < m(∗), and (

e r ⁱ , h

e

r _l ⁱ : l < ωi) ∈ Γ and n(i) = n ^∗ < n ^t for i < i ^∗ < ω and ε ^∗ > 0 and r ∈ Q α

which forces the failure (of (B) of 1.7) for these parameters (the ε ^∗ comes from 1.7). We can assume that r forces

e r ⁱ ∈

e

G Q

for i < i ^∗ (otherwise we can ignore such e

r ⁱ as nothing is demanded on them in 2.11(g)). So r ≥ r ⁱ for i < i ^∗ (see 2.2(F)(β)).

By assumption, for each i < i ^∗ we have: for each r ⁰ ≥ r (hence r ⁰ ≥ r ⁱ and r ⁰ ∈ Random) and i < i ^∗ we have

Av _Ξ

α∗

(ha ⁱ _k (r ⁰ ) : k < ωi) ≥ 1 − h ^t ₁ (n ^∗ ) where (see 2.11(g)( )) we let

a ⁱ _k (r ⁰ ) = a k ( e r, h

e

r l : l < ωi, ¯ n ^t ) = 1 n ^t _k+1 − n ^t _k

X

l∈[n

,n

)

Leb(lim(r ⁰ ) ∩ lim(r ⁱ _l )) Leb(lim(r ⁰ )) . By 1.7 it suffices to prove

Lemma 2.17. Assume Ξ is a finitely additive measure, hB 0 , . . . , B m

−1 i a partition of ω, Ξ(B m ) = a m , i ^∗ < ω and r, r ⁱ _l ∈ Random for i < i ^∗ , l < ω and ¯ n ^∗ = hn ^∗ _i : i < ωi, n ^∗ _i < n ^∗ _i+1 < ω, are such that

(∗) for every r ⁰ ∈ Random, r ⁰ ≥ r and i < i ^∗ we have Av Ξ (ha ⁱ _k (r ⁰ ) : k <

ωi) ≥ b i where

a ⁱ _k (r ⁰ ) = a ⁱ _k (r ⁰ , hr ⁱ _l : l < ωi, ¯ n ^∗ )

= 1

n ^∗ _k+1 − n ^∗ _k

n

−1

X

l=n

Leb(lim(r ⁰ ) ∩ lim(r _l ⁱ )) Leb(lim(r ⁰ )) .

Then for each ε > 0 and k ^∗ < ω there are a finite u ⊆ ω \ k ^∗ and r ⁰ ≥ r such that :

(1) a m − ε < |u ∩ B m |/|u| < a m + ε for m < m ^∗ , (2) for each i < i ^∗ we have

1

|u|

X

k∈u

|{l : n ^∗ _k ≤ l < n ^∗ _k+1 and r ⁰ ≥ r _l ⁱ }|

n ^∗ _k+1 − n ^∗ _k ≥ b _i − ε.

P r o o f. For i < i ^∗ , m < m ^∗ and r ⁰ ≥ r (from Random) let c i,m (r ⁰ ) = Av _ΞB

(ha ⁱ _k (r ⁰ ) : k ∈ B m i) ∈ [0, 1] _R . So clearly

(∗) 1 for r ⁰ ≥ r (in Random) b i ≤ Av _Ξ (ha ⁱ _k (r ⁰ ) : k < ωi)

= X

m<m

Av _ΞB

(ha ⁱ _k (r ⁰ ) : k ∈ B m i)Ξ(B _m ) = X

m<m

c i,m (r ⁰ )a m . There are r ^∗ ≥ r and a sequence ¯ c = hc i,m : i < i ^∗ , m < m ^∗ i such that:

(∗) 2 (a) c i,m ∈ [0, 1] _R , (b) P

m<m

c m,i a m ≥ b _i ,

(c) for every r ⁰ ≥ r ^∗ there is r ⁰⁰ ≥ r ⁰ such that

(∀i < i ^∗ )(∀m < m ^∗ )[c i,m − ε < c _i,m (r ⁰⁰ ) < c i,m + ε].

[Why? Let k ^∗ < ω be such that 1/k ^∗ < ε/(10i ^∗ m ^∗ ) (so k ^∗ > 0). Let Γ = {¯ c : ¯ c = hc i,m : i < i ^∗ , m < m ^∗ i, c _i,m ∈ [0, 1] _R and k ^∗ c i,m is an integer and P

m<m

c i,m a m ≥ b _i } for i < i ^∗ . Clearly Γ is finite and let us list it as h¯ c ^s : s < s ^∗ i. We try to choose by induction on s ≤ s ^∗ a condition r s ∈ Random such that r 0 = r, r s ≤ r s+1 , and if ¯ c ^s satisfies (∗) 2 (a) + (b) then for no r ⁰⁰ ≥ r _s+1 do we have

(∀i < i ^∗ )(∀m < m ^∗ )[c ^s _i,m − ε < c i,m (r ⁰⁰ ) < c ^s _i,m + ε].

For s = 0 we have no problem. If we succeed in arriving at r s

, for i < i ^∗ , m < m ^∗ we can define c ^∗ _i,m ∈ {l/k ^∗ : l ∈ {0, . . . , k ^∗ }} such that c _i,m (r s

) ≤ c ^∗ _i,m < c i,m (r s

) + ε/(10i ^∗ m ^∗ ). By (∗) 1 we have b i ≤ P

m<m

c i,m (r ^∗ _s

)a m . Clearly

X

m<m

c ^s _i,m a m ≥ X

m<m

c i,m (r ^∗ _s

)a m

so ¯ c ^∗ = hc ^∗ _i,m : i < i ^∗ , m < m ^∗ i satisfies (∗) ₂ (a) + (b) and ¯ c ^∗ ∈ Γ , hence

¯

c ^∗ = ¯ c ^s for some s < s ^∗ . But then r ^∗ contradicts the choice of r s+1 . Also by the above Γ 6= ∅. So we are necessarily stuck at some s < s ^∗ , i.e. cannot find r s+1 as required. As ¯ c ⁺ satisfies (∗) 2 (a) + (b), this means that r s , ¯ c ^s as needed in (∗) 2 , so r ^∗ , ¯ c as required exist.]

Let k ^∗ < ω be given. Now choose s ^∗ < ω large enough and try to choose, by induction on s ≤ s ^∗ , a condition r s ∈ Random and natural numbers (m s , k s ) (flipping coins along the way) such that:

(∗) 3 (a) r 0 = r ^∗ , (b) r s+1 ≥ r _s ,

(c) c i,m − ε < c i,m (r s ) < c i,m + ε for i < i ^∗ , m < m ^∗ , (d) k s > k ^∗ , k s+1 > k s ,

(e) k s ∈ B _m

.