THE MEASURE ALGEBRA DOES NOT ALWAYS EMBED

ALAN DOW* AND KLAAS PIETER HART

Abstract. The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(ℕ)/fin. We also discuss the errors in previous results on the embeddability of the measure algebra.

Introduction

The aim of this paper is to prove the following result.

Main Theorem. The Open Colouring Axiom implies that the measure algebra cannot be embedded into the Boolean algebra P(ℕ)/fin.

By ‘the measure algebra’ we mean the quotient of the σ-algebra of Borel sets of the real line by the ideal of sets of measure zero.

There are various reasons, besides sheer curiosity, why it is of interest to know whether the measure algebra can be embedded into P(ℕ)/fin. One reason is that there is great interest in determining what the subalgebras of P(ℕ)/fin are. One of the earliest and most influential result in this direction is Parovičenko’s theorem from [11], which states that every Boolean algebra of size ℵ_{1} can be embedded into P(ℕ)/fin — with the obvious corollary that the Continuum Hypothesis (CH) implies that P(ℕ)/fin is a universal Boolean algebra of size 𝔠: a Boolean algebra embeds into P(ℕ)/fin iff it is of size 𝔠 or less. It is therefore natural to ask how much of this universality remains without assumptions beyond ZFC. It has long been known that every σ-centered Boolean algebra embeds into P(ℕ)/fin but the question for more general c.c.c. Boolean algebras has proven to be much more difficult — with the case of the measure algebra being seen as a touchstone.

This particular case was especially interesting since, by Stone duality, an embedding of the measure into P(ℕ)/fin would provide a c.c.c. nonseparable remainder of ℕ. A ZFC construction of such a remainder was given by Bell in [2]; such remainders were put to good use by Van Mill in [17], see also his survey [18]. The question of the embeddability of the measure algebra remained however.

In recent years many results about maps between Boolean algebras or topological spaces, which were shown to hold under CH, were shown to fail under OCA — see for example [8, 5, 4]. Our result and its proof fall into the same category: in all cases OCA implies that the desired map should have some simple structure whereas one can show, usually in ZFC, that the desired map cannot have this simple structure.

We should note that our result clashes with the result of Frankiewicz and Gutek from [7], which says that Martin’s Axiom implies that the measure algebra can be embedded into P(ℕ)/fin; for completeness we shall point out a gap in their proof. In addition we feel that we should mention the paper [6]. The main result in that paper is that one can establish the consistency of the nonembeddability of the measure algebra using Shelah’s oracle-c.c. method. Regrettably, again the argumentation appears to be incomplete as we shall discuss further in Section 5.

1991 Mathematics Subject Classification. Primary: 28A60. Secondary: 06E99, 03E35, 54G05.

Key words and phrases. measure algebra, embedding, Open Colouring Axiom, P(ℕ)/fin.

*The research of the first author was supported in part by the Netherlands Organization for Scientific Research (NWO) — Grant B 61-408.

The paper is organized as follows. Section 1 contains the necessary preliminaries, including a discussion of what ‘simple structure’ means in our context. Section 2 shows that no embedding can have this simple structure. In Sections 3 and 4 we show how OCA implies that embeddings must have a simple structure. Finally then Section 5 is devoted to the discussion of problems in the previously published work in this area. This will allow us to resurrect the interesting question of whether this result can be established with the oracle-c.c. method.

1. Preliminaries

The Measure Algebra. The standard representation of the Measure Algebra is as the quotient of the σ-algebra of Borel sets of the unit interval by the ideal of sets of Lebesgue measure zero. For ease of notation we choose a different underlying set, namely ℂ = ω×2^{ω}, where 2^{ω}is the Cantor set. We consider the Cantor set endowed with the natural coin-tossing measure μ, determined by specifying μ([s]) = 2^{−|s|}. Here s denotes a finite partial function from ω to 2 and [s] = {x ∈ 2^{ω} : s ⊂ x}. We extend μ on the Borel sets of ℂ by setting μ({n} × [s]) = 2^{−|s|} for all n and s.

The measure algebra is isomorphic to the quotient algebra 𝕄 = Bor(ℂ)/N, where N = {N ⊆ ℂ : μ(N) = 0}; henceforth we shall work with 𝕄.

Liftings of embeddings. Assume φ : 𝕄 → P(ℕ)/fin is an embedding of Boolean algebras and take a lifting Φ : 𝕄 → P(ℕ) of φ; this is a map that chooses a representative Φ(a) of φ(a) for every a in 𝕄.

We shall be working mostly with the restrictions of φ and Φ to the family of (equivalence classes of) open subsets of ℂ and in particular with their restrictions to the canonical base for ℂ, which is

B = {{n} × [s] : n ∈ ω, s ∈ 2^{<ω}}.

To keep our formulas manageable we shall identify B with the set B = ω × 2^{<ω}. We shall also be using layers/strata of B along functions from ω to ω: for f ∈ ^{ω}ω we put B_{f} = {⟨n, s⟩ : n ∈ ω, s ∈ 2^{f(n)}}.

For a subset O of B_{f} we abbreviate φ(⋃{{n} × [s] : ⟨n, s⟩ ∈ O}) by φ(O) and define Φ(O) similarly. Observe that O ↦ φ(O) defines an embedding of P(B_{f}) into P(ℕ)/fin. As an extra piece of notation we use Φ[O] (square brackets) to denote the union ⋃{Φ(n, s) : ⟨n, s⟩ ∈ O}, where Φ(n, s) abbreviates Φ({⟨n, s⟩}).

For later use we explicitly record the following easy lemma.

Lemma 1.1. If f ∈ ^{ω}ω and if O is a finite subset of B_{f} then Φ(O) =^{∗} Φ[O].

Proof. Both sets represent φ(O). □

Let us call a lifting complete if it satisfies Lemma 1.1 for every f ∈ ^{ω}ω and every subset O of B_{f}.

We can always make a lifting Φ exact, by which we mean that the sets Φ(n, ∅) form a partition of ℕ and that every Φ(s, n) is the disjoint union of Φ(n, s^{⌢}0) and Φ(n, s^{⌢}1); indeed, we need only change each of the countably many sets Φ(n, s) by adding or deleting finitely many points to achieve this.

Our proof may now be summarized in a few lines:

(1) For every exact lifting Φ of an embedding φ there are an f ∈ ^{ω}ω and an infinite subset O of B_{f} such that Φ(O) ≠^{∗} Φ[O], i.e., no exact lifting is complete — see Proposition 2.1.

(2) OCA implies that every embedding φ gives rise to an embedding ϕ with a lifting Φ that is both exact and complete — see Section 3.

Together these two statements show that under OCA there cannot be any embedding of 𝕄 into P(ℕ)/fin at all.

The Open Colouring Axiom. The Open Coloring Axiom (OCA) was formulated by Todorčević in [15]. It reads as follows: if X is separable and metrizable and if [X]^{2} = K_{0} ∪ K_{1}, where K_{0} is open in the product topology of [X]^{2}, then either X has an uncountable K_{0}-homogeneous subset Y or X is the union of countably many K_{1}-homogeneous subsets.

One can deduce OCA from the Proper Forcing Axiom (PFA) or prove it consistent in an ω_{2}-length countable support proper iterated forcing construction, using ♢ on ω_{2} to predict all possible subsets of the Hilbert cube and all possible open colourings of these.

We shall not apply OCA directly but use some of its known consequences to prove our main result. A major application occurs in Section 4, where we rely on a result from [4] regarding the behaviour of embeddings of P(ℕ)/fin into itself.

Here and in the next subsection we collect some results of a more general nature. To begin a definition: 𝔟 is the minimum cardinality of a family F of functions from ℕ to ℕ for which there is no upper bound with respect to ⩽^{∗}, i.e., whenever g ∈ ℕ^{ℕ} there if f ∈ F such that {n : f(n) > g(n)} is infinite.

The first consequence of OCA that we need is the equality 𝔟 = ℵ_{2}; it was established in [1, Theorem 3.16]. It follows that the following lemma also holds under OCA.

Lemma 1.2 (𝔟 ⩾ ℵ_{2}). Assume f ↦ α_{f} is a map from ^{ω}ω to ω_{1} that is monotone with respect to ⩽^{∗} and ∈, i.e., if f ⩽^{∗} g then α_{f} ⩽ α_{g}. Then the map is bounded, i.e., there is an α such that α_{f} ⩽ α for all f.

Proof. Because 𝔟 ⩾ ℵ_{2} there is a ⩽^{∗}-cofinal family F in ^{ω}ω on which our map constant, say with value α. Because the map is monotone this α is the ordinal that we are looking for. □

Coherent families of functions. Twice in our proof we shall want to combine a family of partial functions into one single function. In both cases we shall have an ideal I of subsets of some countable set C and for each I a function f_{I} with domain I such that whenever I ⊆ J in I we have f_{J} ↾ I =^{∗} f_{I} — such a family will be called coherent. The following theorem, which is Theorem 3.13 from [1], tells us when a coherent family can be uniformized, i.e., when we can get one function f with domain ⋃ I such that f ↾ I =^{∗} f_{I} for all I.

Theorem 1.3 (OCA). If I is a P_{ℵ_{1}}-ideal then every coherent family of functions on I with values in ω can be uniformized.

An ideal I is a P_{ℵ_{1}}-ideal if for every subfamily I′ of I of size ℵ_{1} (or less) one can find an element J of I such that I ⊆^{∗} J for all I ∈ I′.

We shall need the following generalization of Theorem 1.3 — it actually turns out to be a special case.

Theorem 1.4 (OCA). If I is a P_{ℵ_{1}}-ideal on ω then every coherent family of functions on I with values in P(ℕ) can be uniformized.

Proof. Let {f_{I} : I ∈ I} be a coherent family of functions, with values in P(ℕ). For I ∈ I and g ∈ ^{ω}ω let L_{I,g} = {⟨n, m⟩ : n ∈ I, m ⩽ g(n)} and R_{I,f} = {⟨n, m⟩ : n ∈ I, m ⩽ g(n), m ∈ f_{I}(n)}. The sets L_{I,g} generate a P_{ℵ_{1}}-ideal on the countable set ℕ × ℕ and one readily checks that R_{I,g} =^{∗} R_{J,h} ∩ L_{I,g} whenever I ⊆ J and g <^{∗} h.

Now apply Theorem 1.3 to find R ⊆ ℕ × ℕ such that R ∩ L_{I,g} =^{∗} R_{I,g} for all I and g. This defines f : ω → P(ℕ) by m ∈ f(n) iff ⟨n, m⟩ ∈ R.

If f ↾ I were not almost equal to f_{I} then we’d find infinitely many n with an m_{n} in f(n) △ f_{I}(n). But then R ∩ L_{I,g} ≠^{∗} R_{I,g}, where g ∈ ^{ω}ω follows n ↦ m_{n}. □

2. No exact lifting is complete

Assume φ : 𝕄 → P(ℕ)/fin is an embedding and consider an exact lifting Φ of φ. The following proposition shows that Φ is not complete.

Proposition 2.1. There is a sequence ⟨t_{n} : n ∈ ω⟩ in 2^{<ω} such that for the open set O = ⋃_{n∈ω}{n} × [t_{n}] we have Φ(O) ≠^{∗} Φ[O].

Proof. Take, for each n, the monotone enumeration {k(n, i) : i ∈ ω} of Φ(n, ∅) and apply the equalities to find t(n, i) ∈ 2^{i+2} such that k(n, i) ∈ Φ(n, t(n, i)). Use these t(n, i) to define open sets U_{n} = ⋃_{i∈ω}{n} × [t(n, i)]; observe that μ(U_{n}) ⩽ ∑_{i∈ω} 2^{−i−2} = \frac{1}{2}. It follows that Φ(\{n\} × U_{n}^{c}) is infinite.

We let F be the closed set ⋃_{n∈ω}{n} × U_{n}^{c}; its image Φ(F) meets every Φ(n, ∅) in an infinite set. For every n let i_{n} be the first index with k(n, i_{n}) ∈ Φ(F) and consider the open set O = ⋃_{n∈ω}{n} × [t(n, i_{n})] and the infinite set I = {k(n, i_{n}) : n ∈ ω}.

Observe the following

(1) Φ(O) ∩ Φ(F) =^{∗} ∅, because O ∩ F = ∅;

(2) I ⊆ Φ(F), by our choice of the i_{n}; and

(3) I ⊆ Φ[O], by the choice of the t(n, i_{n}). It follows that ⟨t(n, i_{n}) : n ∈ ω⟩ is as required. □

This completes the proof of the first half of the main argument.

3. OCA gives embeddings with exact liftings that are complete

We assume φ is an embedding of 𝕄 into P(ℕ)/fin. We shall find, assuming OCA, an infinite set A and a lifting Φ of φ that is exact and complete when restricted to subsets of A × 2^{<ω}.

The infinite set A will come from an almost disjoint family on ω: we fix bijection π between ω and 2^{<ω} and define for x ∈ 2^{ω} the set A_{x} by A_{x} = π^{←}[{x ↾ n : n ∈ ω}].

For the rest of this section we fix an ℵ_{1}-sized subfamily A of the A_{x}’s and enumerate it as {A_{α} : α < ω_{1}}. Using OCA we shall show that all but countably many A_{α} are as required.

By construction the family A has a special property, commonly referred to as neatness; an almost disjoint family C is neat if there is a bijection π between ω and 2^{<ω} such that for every C ∈ C the set π[C] is a subset of some branch x_{C} and, moreover, the map C ↦ x_{C} is one-to-one.

Our final lifting will be a limit, via Theorem 1.4, of a coherent family of liftings; these liftings will be defined only partially so we fix an exact lifting Ψ of φ to extend these partial liftings.

The key technical result is the following; we postpone its proof until Section 4.

Theorem 3.1 (OCA). Let φ be an embedding of P(ℕ) into P(ℕ)/fin and let A be a neat almost disjoint family on ℕ of size ℵ_{1}. Then for all but countably many A ∈ A there are D ⊆ ℕ and a function H : D → A such that φ(x) = H^{←}[x]^{∗} for all x ⊆ A.

This theorem will now be applied to establish the following proposition.

Proposition 3.2 (OCA). For every f ∈ ^{ω}ω there is a β < ω_{1} such that for every α ⩾ β there is a lifting Φ_{f,α} of φ with Φ_{f,α}(O) = Φ_{f,α}[O] whenever O ⊆ B_{f,A_{α}} and such that Φ_{f,α}(n, s) ∩ Φ_{f,α}(m, t) = ∅ whenever {n} × [s] and {m} × [t] are disjoint.

Proof. We fix f ∈ ^{ω}ω and show how to find β and Φ_{f,α} for each α ⩾ β. We transfer the almost disjoint family A to B_{f} by setting C_{α} = B_{f,A_{α}} and C = {C_{α} : α < ω_{1}}.

It is fairly straightforward to show that C is neat; one stretches the bijection π to find an injection π̃ from B_{f} to 2^{<ω} that maps every C_{α} onto a branch of 2^{<ω} and different C_{α} to different branches.

This means that we can apply Theorem 3.1, to be proved in the next section, to the embedding φ_{f} of P(B_{f}) into P(ℕ)/fin, defined by φ_{f}(O) = φ(O). This gives us a β and for every α ⩾ β a subset D_{α} of ℕ and a function H_{α} : D_{α} → C_{α} such that for every subset O of B_{f} the set H\overleftarrow{α}[O] is a representative of φ_{f}(O). We can define Φ_{f,α} by Φ_{f,α}(O) = H^{←}[O] for O ⊆ B_{f} and by setting Φ_{f,α}(a) = Ψ(a) for the other elements of 𝕄. □

For each f we denote the minimum possible β by α_{f}.

Lemma 3.3. If f ⩽^{∗} g then α_{f} ⩽ α_{g}.

Proof. Let α ⩾ α_{g} and consider the lifting Φ_{g,α}. We define a lifting Φ_{f,α} in a fairly obvious way: first fix m such that f(n) ⩽ g(n) for n ⩾ m and, if need be, redefine, for the duration of this proof, the values Ψ(n, s) for n < m and s ∈ 2^{f(n)} so as to get Ψ(n, s) ∩ Φ_{g,α}(l, t) = ∅ whenever this is needed.

Then, given O ⊆ B_{f,A_{α}} put O_{1} = {⟨n, s⟩ ∈ O : n ⩾ m} and put

U_{O} = {⟨n, t⟩ ∈ B_{g,A_{α}} : (∃⟨n, s⟩ ∈ O_{1})(s ⊆ t)}.

We define, for O ⊆ B_{f,A_{α}},

Φ_{f,α}(O) = Φ_{g,α}[U_{O}] ∪ Ψ[{⟨n, s⟩ ∈ O : n < m}].

Note that we implicitly defined Φ_{f,α}(n, s) = Φ[{⟨n, t⟩ ∈ B_{g} : s ⊆ t}] whenever n ⩾ m and Φ_{f,α}(n, s) = Ψ(n, s) when n < m. It follows that Φ_{f,α}(O_{1}) = Φ_{g,α}[U_{O}] = Φ_{f,α}[O_{1}] and hence that

Φ_{f,α}(O) = Φ_{f,α}[O].

We already took care of the disjointness requirement so this Φ_{f,α} witnesses that α ⩾ α_{f} (once we use Ψ to define Φ_{f,α} on the rest of 𝕄). □

We apply Lemma 1.2 to find α_{∞} such that α_{f} ⩽ α_{∞} for all f.

We fix α ⩾ α_{∞} and put A = A_{α}. For every f ∈ F we simply write Φ_{f} for the lifting Φ_{f,α}. For every f we extend Φ_{f} in a natural way to the set C_{f} = {⟨n, s⟩ : n ∈ A, |s| ⩽ f(n)}: we demand that Φ_{f}(s, n) = Φ_{f}(n, s^{⌢}0) ∪ Φ_{f}(n, s^{⌢}1) whenever appropriate; this makes Φ_{f} exact on C_{f}.

Lemma 3.4. If f ⩽^{∗} g then Φ_{g} ↾ C_{f} =^{∗} Φ_{f}.

Proof. Consider a potential sequence ⟨⟨n_{i}, s_{i}⟩ : i ∈ ω⟩ of points in C_{f} where Φ_{g} and Φ_{f} disagree. By the disjointness condition and because the symmetric difference of Φ_{f}(n_{i}, s_{i}) and Φ_{g}(n_{i}, s_{i}) is always finite we can assume that Φ_{f}(n_{i}, s_{i}) does not meet Φ_{g}(n_{j}, s_{j}) when i < j. Let O_{g} = {⟨n_{i}, s⟩ ∈ B_{g} : i ∈ ω, s_{i} ⊆ s} and O_{f} = {⟨n_{i}, s⟩ ∈ B_{f} : i ∈ ω, s_{i} ⊆ s}.

Observe that O_{g} and O_{f} determine the same open subset of A × 2^{ω}, so that Φ_{g}(O_{g}) =^{∗} Φ_{f}(O_{g}). It should be clear however that by the choice of the points ⟨n_{i}, s_{i}⟩ we have Φ_{g}[O_{g}] ≠^{∗} Φ_{f}[O_{f}], which is a contradiction. □

Observe that because 𝔟 = ℵ_{2} the family {C_{f} : f ∈ ^{ω}ω} is a P_{ℵ_{1}}-ideal on A×2^{<ω}; we can therefore apply Theorem 1.4 to find one map Φ from A × 2^{<ω} to P(ℕ) such that Φ ↾ C_{f} =^{∗} Φ_{f} for all f ∈ ^{ω}ω. This function Φ is almost as required.

First choose m ∈ ω and a ⩽^{∗}-cofinal family F consisting of increasing elements of ^{ω}ω such that Φ(n, s) = Φ_{f}(n, s) whenever f ∈ F, n ⩾ m and s ∈ 2^{f(n)}. Without loss of generality the set {f(m) : f ∈ F} is unbounded — make m a bit larger if necessary (if no larger m can be found the family F is not even ⩽^{∗}-unbounded).

This immediately implies that Φ is exact on (A \ m) × 2^{<ω}; we simply modify Φ slightly on (A ∩ m) × 2^{<ω} to make it exact on the whole of A × 2^{<ω}. For all other elements a of 𝕄 we put Φ(a) = Ψ(a).

The proof that Φ is complete is much like the proof of Lemma 3.3. Let f ∈ ^{ω}ω and l ⩾ m such that Φ(n, s) = Φ_{f}(n, s) whenever n ⩽ l. Given O ⊆ B_{f} we first note that Φ(O) =^{∗} Φ_{f}(O), because both Φ and Φ_{f} are liftings.

To complete the proof we show that also Φ[O] =^{∗} Φ_{f}[O]. Indeed, let O′ = {⟨n, s⟩ ∈ O : n ⩾ l}, then Φ[O′] = Φ_{f}[O′], so we are left with showing Φ[O′′] =^{∗} Φ_{f}[O′′], where O′′ = O \ O′. But O′′ is finite so that by Lemma 1.1 we have Φ[O′′] =^{∗} Φ(O′′) and Φ_{f}(O′′) =^{∗} Φ_{f}[O′′]; the equality Φ(O′′) =^{∗} Φ_{f}(O′′) holds because both maps are liftings.

4. Embedding P(ℕ) into P(ℕ)/fin

In this section prove Theorem 3.1, thus completing the argument for our main result. We are given an an embedding φ of P(ℕ) and a neat almost disjoint family A = {A_{α} : α < ω_{1}}. We have to find an α such that for every β ⩾ α there are D ⊆ ℕ and H : D → A_{α} such that φ(x) = H^{←}[x]^{∗} for all subsets x of A_{α}.

We begin by taking a lifting Φ : P(ℕ) → P(ℕ) of φ. We may assume, upon replacing Φ({n}) by {n} ∪ Φ({n}) \ ⋃_{i<n} Φ({i}), that the Φ({n}) form a partition of ℕ. We shall identify ℕ with ℕ × ℕ in such a way that Φ({n}) corresponds to the vertical line {n} × ℕ; we shall therefore write V_{n} for Φ({n}).

For an f ∈ ^{ω}ω we write L_{f} = {⟨n, m⟩ : n ∈ ω, m ⩽ f(n)}. The following lemma will be useful toward the end of the proof.

Lemma 4.1. For each a ⊆ ℕ there is f ∈ ^{ω}ω such that Φ(a) \ L_{f} = ⋃_{n∈a} V_{n} \ L_{f}.

Proof. If n ∈ a then V_{n} ⊆^{∗} Φ(a) and if n ∉ a then V_{n} ∩ Φ(a) =^{∗} ∅; now let f code witnesses: if n ∈ a then V_{n} \ L_{f} ⊆ Φ(a) and if n ∉ a then V_{n} \ L_{f} ∩ Φ(a) = ∅. □

We enumerate A as {A_{α} : α ∈ ω_{1}}.

For f ∈ ^{ω}ω consider Φ_{f} : P(ℕ) → P(L_{f}), defined by Φ_{f}(a) = Φ(a) ∩ L_{f} and observe that Φ_{f} induces a homomorphism from P(ℕ)/fin to P(ℕ)/fin.

As in [4] OCA may now be applied to give us an α_{f} < ω_{1} such that Φ_{f} is simple on A_{α} whenever α ⩾ α_{f}, where ‘simple’ means that there are D ⊆ L_{f} and a finiteto-one function h : D → A_{α} such that Φ_{f}(a) =^{∗} h^{←}[a] for all subsets a of A_{α}. As in the previous section we choose α_{f} as small as possible and we use the following lemma to fix α_{∞} such that α_{f} ⩽ α_{∞} for all f.

Lemma 4.2. If f ⩽^{∗} g then α_{f} ⩽ α_{g}.

Proof. Take α ⩾ α_{g} and fix D ⊆ L_{g} and h : D → A_{α} such that Φ_{g}(a) =^{∗} h^{←}[a] for all a ⊆ A_{α}. Now simply let D_{1} = D ∩ L_{f} and h_{1} = h ↾ D_{1}; clearly Φ_{f}(a) =^{∗} Φ_{g} ∩ L_{f} =^{∗} h^{←}[a] ∩ L_{f} = h\overleftarrow{1}[a] for all a ⊆ A_{α}. We see that α ⩾ α_{f}. □

For the rest of the proof we fix an α ⩾ α_{∞} and show that A = A_{α} is as required. For each f ∈ ^{ω}ω we take D_{f} and h_{f} : D_{f} → A as above. We intend to find D and H by an application of Theorem 1.3.

Lemma 4.3. If f ⩽ g then D_{f} =^{∗} D_{g} ∩ L_{f} and h_{g} ↾ D_{f} =^{∗} h_{f}.

Proof. The first equality is clear: by construction D_{g} =^{∗} Φ_{g}(A) and D_{f} =^{∗} Φ_{f}(A), so that D_{g} ∩ L_{f} =^{∗} Φ_{g}(A) ∩ L_{f} =^{∗} Φ_{f}(A) =^{∗} D_{f}.

To prove the second equality let x be an infinite subset of D_{g} ∩ D_{f} such that h_{f}(i) ≠ h_{g}(i) for all i ∈ x; because h_{f} and h_{g} are finite-to-one we can assume that h_{f}[x] ∩ h_{g}[x] = ∅. But then we would have a contradiction because on the one hand Φ_{g}(h_{f}[x])∩Φ_{g}(h_{g}[x]) =^{∗} h\overleftarrow{g}[h_{f}[x]]∩h\overleftarrow{g}[h_{g}[x]] = ∅ while on the other hand x ⊆ h\overleftarrow{f}[h_{f}[x]] ∩ h\overleftarrow{g}[h_{g}[x]] ⊆^{∗} Φ_{g}(h_{f}[x]) ∩ Φ_{g}(h_{g}[x]). □

We apply Theorem 1.3 to the family {F_{f} : f ∈ ^{ω}ω} of functions defined by F_{f}(p) = ⟨χ_{D_{f}} (p), h_{f}(p)⟩ to find a function F : ω × ω → 2 × ω that uniformizes this family. We set D = {p : F_{1}(p) = 1} and H = F_{2} ↾ D.

For the rest of the proof the letters n and m will refer to elements of A.

Lemma 4.4. There are only finitely many pairs ⟨n, m⟩ for which H^{←}(n) ∩ V_{m} is infinite.

Proof. Assume we have {⟨n_{i}, m_{i}⟩ : i ∈ ω} with H^{←}(n_{i}) ∩ V_{m_{i}} infinite for all i and n_{i}, m_{i} < n_{j}, m_{j} whenever i < j. Choose f ∈ ^{ω}ω as per Lemma 4.1 for the sets b = {n_{i} : i ∈ ω} and c = {m_{i} : i ∈ ω} and choose g > f such that H^{←}(n_{i}) ∩ V_{m_{i}} ∩ (L_{g} \ L_{f}) ≠ ∅ for all i.

We obtain a contradiction as before: b and c are disjoint, hence Φ(b) and Φ(c) are almost disjoint. On the other hand h\overleftarrow{g}[b] \ L_{f} ⊆^{∗} Φ(b) and ⋃_{m∈c}\{m\} × (f(m), g(m)] ⊆ Φ(c); the intersection of the smaller sets is infinite. □

Lemma 4.5. There are only finitely many m in A for which the set {n ∈ A : H^{←}(n) ∩ V_{m} ≠ ∅} is infinite.

Proof. Let b be the set of m in A for which I_{m} = {n ∈ A : H^{←}(n) ∩ V_{m} ≠ ∅} is infinite. Thin out b to get I_{m} \ b infinite for all m in b. Choose f ∈ ^{ω}ω as per Lemma 4.1 for A and b, so that Φ(b) \ L_{f} = (b × ω) \ L_{f} and Φ(A \ b) \ L_{f} = ((A \ b) × ω) \ L_{f}.

Because the sets H^{←}(n) are pairwise disjoint we can a one-to-one choice function m ↦ n_{m} for the family {I_{m} \ b : m ∈ b} such that H^{←}(n_{m}) ∩ V_{m} \ L_{f} ≠ ∅ for all m.

We choose a function g > f that captures these intersections: H^{←}(n_{m}) ∩ V_{n} ∩ (L_{g} \ L_{f}) ≠ ∅ for all m. It follows that H^{←}[b] ∩ (L_{g} \ L_{f}) meets (A \ b) × ω in an infinite set. However, by the choice of f and the properties of h_{g} the set H^{←}[b] ∩ (L_{g} \ L_{f}) is almost equal to ⋃_{m∈b}{m} × (f(m), g(m)] which is disjoint from (A \ b) × ω. □

Lemma 4.6. For every n ∈ A the set {m ∈ A : H^{←}(n) ∩ V_{m} ≠ ∅} is finite.

Proof. Fix f ∈ ^{ω}ω such that for all n and m in A: if n ≠ m and H^{←}(n) ∩ V_{m} ≠ ∅ then H^{←}(n) ∩ V_{m} ∩ L_{f} ≠ ∅. Now note that H^{←}(n) ∩ L_{f} =^{∗} h\overleftarrow{f}(n), so that H^{←}(n) ∩ L_{f} is finite. □

Putting these lemmas together we see that there are M and N in ω such that N ⩾ M and

(1) if n, m ⩾ M and n ≠ m then H^{←}(n) ∩ V_{m} is finite;

(2) if m ⩾ M then V_{m} meets only finitely many H^{←}(n); and

(3) if n < M and m ⩾ N then H^{←}(n) ∩ V_{m} = ∅. (Note that M should be chosen first, to ensure 1 and 2.)

By 1 and 2 we can fix h ∈ ^{ω}ω such that H^{←}(n) ∩ V_{m} ⊆ L_{h} whenever n, m ⩾ M and n ≠ m; an application of 3 then tells us that H^{←}(n) \ L_{h} = V_{n} \ L_{h} for n ⩾ N. We also redefine H on the set N × ω to get H^{←}(n) = V_{n} for n < N.

Now let b ⊆ A and fix f > h as per Lemma 4.1. By the choice of f and h we have

Φ(b) \ L_{f} = (b × ω) \ L_{f} = H^{←}[b] \ L_{f}.

The redefined H still satisfies H ↾ L_{f} =^{∗} h_{f} so that

Φ(b) ∩ L_{f} =^{∗} h\overleftarrow{f}[b] =^{∗} H^{←}[b] ∩ L_{f}.

This shows that H is as required.

5. Comments and questions

In this section we comment on two earlier results about the embeddability of 𝕄 into P(ℕ)/fin and raise some questions.

5.1. Martin’s Axiom does not imply embeddability. A consequence of our result is that Martin’s Axiom does not imply that 𝕄 can be embedded into P(ℕ)/fin; this is so because the conjunction OCA + MA is consistent — it follows from the Proper Forcing Axiom and it can be proved consistent in an ω_{2}-length iterated forcing construction.

In [7] Frankiewicz and Gutek assert that MA implies there is a measure-preserving embedding φ of 𝕄 into P(ℕ)/fin — measure preserving in the sense that for every element a one has μ(a) = d(φ(a)). Here d is the asymptotic density, defined by

d(X) = lim_{n→∞} \frac{|X ∩ \{1, 2, …, n\}|}{n},

for those subsets X of ℕ for which the limit exists. Of course for this to make sense we must consider the standard incarnation of 𝕄 as the quotient of the Borel algebra of the unit interval by the ideal of measure-zero sets.

The reader is likely to be interested where the argument has a gap. It seems that the principal error in their proof is in the following lemma, which is the key step in the construction of the embedding.

Lemma 5.1 (MA). If 𝕃 is a subalgebra of 𝕄 of size less than 𝔠 and if a ∈ 𝕄 \ 𝕃 then every measure-preserving embedding φ of 𝕃 into P(ℕ)/fin can be extended to a measure-preserving embedding ψ of the algebra generated by 𝕃 ∪ {a} into P(ℕ)/fin.

It is relatively easy to see that this lemma is true, in ZFC, for countable 𝕃 — indeed, a value ψ(a) is readily constructed by recursion. This lemma is false for 𝕃 of size ℵ_{1}, as can be seen from the following example. We work on the interval (0, 1]. We split (0, 1] into intervals {I_{n} : n ∈ ℕ}, where I_{2n−1} = (2^{−(n+1)}, 2^{−n}] and I_{2n} = I_{2n−1} + \frac{1}{2}. Next let A = \{A_{α} : α < ω_{1}\} be an ℵ_{1}-sized almost disjoint family, where A_{α} consists of even/odd numbers whenever α is even/odd. For α < ω_{1} we put a_{α} = ⋃_{n∈A_{α}} I_{n}; note that {a_{α} : α < ω_{1}} is an independent family in 𝕄 in that no element is in the algebra generated by the other elements of the family — we let 𝕃 be the subalgebra of 𝕄 generated by this family. Observe that a = (0, \frac{1}{2}] does not belong to 𝕃.

Let L = {L_{α} : α < ω_{1}} be a Luzin-type almost disjoint family, which means that for no two disjoint uncountable subsets S and T of ω_{1} one can find a subset X of ℕ such that L_{α} ⊆^{∗} X for α ∈ S and L_{α} ∩ X =^{∗} ∅ for α ∈ T — see [10] or [16, Theorem 4.1] . We assume the family L lives on the set Z = {n! : n ∈ ℕ} — note that d(Z) = 0.

We can apply the countable version of Lemma 5.1 to find a measure-preserving embedding from the algebra 𝕃 into P(ℕ \ Z)/fin. Now augment this embedding to obtain an embedding φ of 𝕃 into P(ℕ)/fin such that φ(a_{α}) ∩ Z = L_{α} for all α. There is no way to extend φ to an embedding of 𝕃 ∪ {a} into P(ℕ)/fin, measurepreserving or otherwise: because a_{α} < a if α is odd and a_{α} ∧ a = 0 if α is even we should have L_{α} ⊆^{∗} ψ(a) for odd α and L_{α} ∩ ψ(a) =^{∗} ∅ for even α, which is impossible by the choice of L.

5.2. The oracle-c.c. method. In [6] Frankiewicz sketched a proof of the consistency of the nonembeddability of 𝕄, using Shelah’s oracle-c.c. method. Certainly this sketch is quite incomplete but it also appears to be fundamentally erroneous. Before we can point out where Frankiewicz went wrong we give a rough outline of the oracle-c.c. method.

Shelah’s oracle-c.c. method — see [3], [12, Chapter IV] and [14, Chapter IV] — provides us with a way of preventing that various undesirable objects will appear once an iterated forcing construction is underway. It would take us too far afield to explain this method in full, suffice it to say that one builds c.c.c. partial orders of size ℵ_{1} whose antichains are kept under tight control by a ♢-sequence (an oracle).

A consistency proof, along these lines, for the nonembeddability of 𝕄 would run as follows: given an embedding φ : 𝕄 → P(ℕ)/fin, construct a partial order ℙ and a ℙ-name Ẋ for a Borel set such that there is no ℙ ∗ Fn(ω, 2)-name Ȧ for a subset of ℕ that satisfies the following two requirements for every Borel set Y from the ground model:

⊩ “ Y ⊆ Ẋ ⇒ φ(Y) ⊆^{∗} Ȧ ” and ⊩ “ Ẋ ⊆ Y ⇒ Ȧ ⊆^{∗} φ(Y) ”

The extra factor Fn(ω, 2) is a necessary technical device (implicit in any finitesupport iteration) that will enable one to improve a given ♢-sequence into an oracle with the property that if the rest of the iteration is kept under its control one will never encounter an undesirable name Ȧ as above. This, together with a reflection argument involving ♢ on ω_{2}, will ensure that ultimately no embedding of 𝕄 into P(ℕ)/fin will remain.

Frankiewicz’ sketch is based on Shelah’s proof, from [13], that it is consistent that the natural quotient homomorphism q from Bor(ℂ) onto 𝕄 does not split, i.e., there is no homomorphism h : 𝕄 → Bor(ℂ) such that q ∘ h = Id_{𝕄} (such a homomorphism is also called a lifting of 𝕄 into Bor(ℂ)).

The sketch consists basically of two parts.

(1) A copy of Shelah’s construction of the partial order from [13] with some (questionable) modifications, and

(2) the unsupported assertion that “From now on the proof goes exactly as in

Shelah [13]”. The problem with the second part is that Shelah’s argument was written up in such a way that it would also apply to the category algebra; this is the quotient algebra 𝕂 = Bor(ℂ)/M, where M is the ideal of meager sets. As none of the modifications is 𝕄-specific Frankiewicz’ sketch would also lead to a proof that it is consistent for there to be no embedding of the category algebra into P(ℕ)/fin. This, however, is impossible: the category algebra is known to be embeddable into P(ℕ)/fin. This is most readily seen by noting its Stone space is separable and hence a continuous image of βℕ \ ℕ.

One very problematic modification occurs almost at the beginning: we are given a countable partial order ℙ and a ℙ∗Fn(ω, 2)-name Ȧ for a subset of ℕ. We are then promised an extension ℙ′ of ℙ and an infinite subset B of ℕ such that some condition in ℙ′ ∗ Fn(ω, 2) will force either B ⊆ Ȧ or B ∩ Ȧ = ∅. This is patently impossible if Ȧ happens to be the name of the generic subset that is added by Fn(ω, 2): it is well-known that this set and its complement meet every infinite subset of ℕ from the ground model.

5.3. Some questions. It would be of interest to know whether there are other ways of proving the nonembeddability of the measure algebra consistent. To be specific we ask.

Question 5.1. Can the consistency of the nonembeddability of 𝕄 be established by the oracle-c.c. method?

Because the oracle-c.c. method relies on having ♢ in every intermediate model it seems to produce models with 𝔠 ⩽ ℵ_{2} only; the known models for OCA satisfy 𝔠 = ℵ_{2} as well. Therefore the following question ‘really’ asks for a new method.

Question 5.2. Is the nonembeddability of 𝕄 consistent with larger values for 𝔠?

Many statements that were proved consistent by the oracle-c.c. method were later shown to follow from OCA (or OCA + MA), see, e.g., [8]. One of the major questions that remains is.

Question 5.3. Does OCA or even PFA imply that the quotient homomorphism from Bor(ℂ) onto 𝕄 (or onto 𝕂) does not split?

Another question is to delineate what the subalgebras of P(ℕ)/fin (or, dually, the zero-dimensional continuous images of βℕ \ ℕ) look like. The ZFC results can, broadly, be divided into two categories: the first contains the easy result that every separable compact space is a continuous image of βℕ\ℕ; the second contains moderately difficult constructions of embeddings into P(ℕ)/fin of assorted algebras — these are generally small in some sense or another, so that the construction can be seen to terminate. The smallness conditions all seem to imply that the algebra in question is incomplete. This suggests that the following question might have a positive answer.

Question 5.4. Is it consistent that every complete Boolean algebra that is embeddable into P(ℕ)/fin must be σ-centered?

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Department of Mathematics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3

E-mail address: dowa@YorkU.CA

Faculty of Information Technology and Systems, TU Delft, Postbus 5031, 2600 GA Delft, the Netherlands

E-mail address: k.p.hart@twi.tudelft.nl
URL: http://aw.twi.tudelft.nl/~hart