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Version of 23.10.14/17.9.16 Chapter 54 Real-Valued-Measurable Version of 23.10.14/17.9.16 Chapter 54 Real-valued-measurable cardinals Of the many questions in measure theory which involve non-trivial set theory, perhaps the first to have been recognised as fundamental is what I call the ‘Banach-Ulam problem’: is there a non-trivial measure space in which every set is measurable? In various forms, this question has arisen repeatedly in earlier volumes of this treatise (232Hc, 363S, 438A). The time has now come for an account of the developments of the last fifty years. The measure theory of this chapter will begin in 543; the first two sections deal with generalizations to § wider contexts. If ν is a probability measure with domain X, its null ideal is ω1-additive and ω1-saturated in X. In 541 I look at ideals ⊳ X such that is simultaneouslyP κ-additive and κ-saturated for some κ; thisP is already§ enough to lead usI toP a version of theI Keisler-Tarski theorem on normal ideals (541J), a great strengthening of Ulam’s theorem on inaccessibility of real-valued-measurable cardinals (541Lc), a form of Ulam’s dichotomy (541P), and some very striking infinitary combinatorics (541Q-541S). In 542 I specialize § to the case κ = ω1, still without calling on the special properties of null ideals, with more combinatorics (542E, 542I). I have said many times in the course of this treatise that almost the first thing to ask about any measure is, what does its measure algebra look like? For an atomless probability measure with domain X, the Gitik-Shelah theorem (543E-543F) gives a great deal of information, associated with a tantalizingP problem (543Z). 544 is devoted to the measure-theoretic consequences of assuming that there is some atomlessly- measurable§ cardinal, with results on repeated integration (544C, 544I, 544J), the null ideal of a normal witnessing probability (544E-544F) and regressive functions (544M). I do not discuss consistency questions in this chapter (I will touch on some of them in Chapter 55). The ideas of 541-544 would be in danger of becoming irrelevant if it turned out that there can be no two- valued-measurable§§ cardinal. I have no real qualms about this. One of my reasons for confidence is the fact that very much stronger assumptions have been investigated without any hint of catastrophe. Two of these, the ‘product measure extension axiom’ and the ‘normal measure axiom’ are mentioned in 545. One way of looking at the Gitik-Shelah theorem is to say that if X is a set and is a proper§ σ-ideal of subsets of X, then X/ cannot be an atomless measurable algebra of small MaharamI type. We can ask whether there are furtherP I theorems of this kind provable in ZFC. Two such results are in 547: the ‘Gitik- Shelah theorem for category’ (547F-547G), showing that X/ cannot be isomorphic to RO(§ R), and 547R, showing that ‘σ-measurable’ algebras of moderate complexityP I also cannot appear as power set σ-quotient algebras. This leads us to a striking fact about free sets for relations with countable equivalence classes (548C) and thence to disjoint refinements of sequences of sets (548E). Version of 10.12.12 541 Saturated ideals If ν is a totally finite measure with domain X and null ideal (ν), then its measure algebra X/ (ν) P N P N is ccc, that is to say, sat( X/ (ν)) ω1; while the additivity of (ν) is at least ω1. It turns out that an ideal of X such thatP sat( NX/ )≤ add is either trivial or extraordinary.N In this section I present a little ofI theP theory of such ideals.P ToI begin≤ with,I the quotient algebra has to be Dedekind complete (541B). Further elementary ideas are in 541C (based on a method already used in 525) and 541D-541E. In a less § expected direction, we have a useful fact concerning transversal numbers TrI (X; Y ) (541F). The most remarkable properties of saturated ideals arise because of their connexions with ‘normal’ ideals (541G). These ideals share the properties of non-stationary ideals (541H-541I). If is an (add )-saturated I I Extract from Measure Theory, by D.H.Fremlin, University of Essex, Colchester. This material is copyright. It is issued under the terms of the Design Science License as published in http://dsl.org/copyleft/dsl.txt. This is a de- velopment version and the source files are not permanently archived, but current versions are normally accessible through https://www1.essex.ac.uk/maths/people/fremlin/mt.htm. For further information contact [email protected]. c 2005 D. H. Fremlin c 2004 D. H. Fremlin 1 2 Real-valued-measurable cardinals §541 intro. ideal of X, we have corresponding normal ideals on add (541J). Now there can be a κ-saturated normal ideal onPκ only if there is a great complexity of cardinals belowI κ (541L). The original expression of these ideas (Keisler & Tarski 64) concerned ‘two-valued-measurable’ cardi- nals, on which we have normal ultrafilters (541M). The dichotomy of Ulam 1930 (438Ce-438Cf) reappears in the context of κ-saturated normal ideals (541P). For κ-saturated ideals, ‘normality’ implies some far-reaching extensions (541Q). Finally, I include a technical lemma concerning the covering numbers covSh(α,β,γ,δ) (541S). 541A Definition If A is a Boolean algebra, I is an ideal of A and κ is a cardinal, I will say that I is κ-saturated in A if κ sat(A/I); that is, if for every family a in A I there are distinct ξ, η<κ ≥ h ξiξ<κ \ such that a ∩ a / I. ξ η ∈ 541B Proposition Let A be a Dedekind complete Boolean algebra and I an ideal of A which is (add I)+- saturated in A. Then the quotient algebra A/I is Dedekind complete. proof Take any B A/I. Let C be the set of those c A/I such that either c ∩ b = 0 for every b B ⊆ ∈ ∈ or there is a b B such that c ⊆ b. Then C is order-dense in A/I, so there is a partition of unity D C (313K). Enumerate∈ D as d , where ⊆ h ξiξ<κ κ = #(D) < sat(A/I) (add I)+. ≤ • For each ξ<κ, choose aξ A such that aξ = dξ. Now #(ξ) < κ add I, so supη<ξ aξ ∩ aη I. Set • ∈ • ≤ ∈ a˜ξ = aξ \ supη<ξ aη; thena ˜ξ = aξ = dξ. Set L = ξ : ξ<κ, dξ ⊆ b for some b B and a = supξ∈L a˜ξ in A. • { • ∈ } ??? If b B and b ⊆ a , there must be a ξ<κ such that dξ ∩ (b a ) = 0. But dξ C, so there must be a ′ ∈ 6 ′ • \ 6 • ∈ b B such that dξ ⊆ b ; accordingly ξ L,a ˜ξ ⊆ a and dξ ⊆ a . XXX Thus a is an upper bound of B in A/I. ∈ ∗ ∈ • ∗ • ∗ ??? If there is an upper bound b of B such that a ⊆ b , there must be a ξ<κ such that dξ ∩ a \ b = 0. ∗ 6 6 As dξ ⊆ b , dξ ⊆ b for every b B, and ξ / L. But this means thata ˜ξ ∩ a˜η = 0 for every η L, soa ˜ξ ∩ a = 0 6 6 • ∈ ∈ ∈ (313Ba) and dξ ∩ a = 0. XXX Thus a• = sup B in A/I; as B is arbitrary, A/I is Dedekind complete. 541C Proposition Let X be a set, κ a regular infinite cardinal, Σ an algebra of subsets of X such that Σ whenever Σ and #( ) <κ, and a κ-saturated κ-additive ideal of Σ. E ∈ E ⊆ E I S (a) If Σ there is an ′ [ ]<κ such that E ′ for every E . E ⊆ E ∈ E \ E ∈I ∈E (b) If Eξ ξ<κ is any family in Σ , and θ<κSis a cardinal, then x : x X, #( ξ : x Eξ ) θ includes ah memberi of Σ (and, in particular,\I is not empty). { ∈ { ∈ } ≥ } (c) Suppose that no element\I of Σ can be covered by κ members of . Then κ is a precaliber of Σ/ . \I I I proof Write A for Σ/ . I (a) Consider A = E• : E A. By 514Db, there is an ′ [ ]<sat(A) such that E• : E ′ has the same upper bounds{ as ∈A. E} Now ⊆ #( ′) < sat(A) κ, so F =E ∈ ′Ebelongs to Σ, and F{ • must∈ be E an} E ≤ E upper bound for A, that is, E F for every E . S \ ∈I ∈E (b) For α β<κ set F = E Σ. Then for every α<κ we have a g(α) < κ such that ≤ αβ α≤ξ<β ξ ∈ g(α) α and Eξ Fα,g(α) wheneverS g(α) ξ<κ, by (a) and the regularity of κ. Define≥ h(α) \ by setting∈I h(0) = 0, h(α≤+1) = g(h(α)) for each α<κ, and h(α) = sup h(β) for h iα<κ β<α non-zero limit ordinals α<κ. Set Gα = Fh(α),h(α+1) for each α. If β<α, then G G = E F α \ β h(α)≤ξ<h(α+1) ξ \ h(β),g(h(β)) ∈I because is κ-additive. Consequently GS G belongs to ; and G E / , so G = G I θ \ β<θ β I θ ⊇ h(θ) ∈I β<θ β ∈ Σ . But if x G then ξ : x E meetsT [h(β),h(β + 1)[ for every β<θ and has cardinal atT least θ. \I ∈ { ∈ ξ} (c) Let a be a family of non-zero elements in A.
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