Sakae Fuchino, Noam Greenberg and Saharon Shelah- Models of Real-Valued Measurability

8/3/2019 Sakae Fuchino, Noam Greenberg and Saharon Shelah- Models of Real-Valued Measurability

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MODELS OF REAL-VALUED MEASURABILITY

SAKAE FUCHINO, NOAM GREENBERG, AND SAHARON SHELAH

Abstract. Solovays random-real forcing ([1]) is the standard way of pro-ducing real-valued measurable cardinals. Following questions of Fremlin, bygiving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovays model that do not follow from the existenceof real-valued measurability.

1. Introduction

Solovay ([1]) showed how to produce a real-valued measurable cardinal by addingrandom reals to a ground model which contains a measurable cardinal. (Recall thata cardinal is real-valued measurable if there is an atomless, -additive measure on that measures all subsets of . For a survey of real-valued measurable cardinalssee Fremlin [2].)

The existence of real-valued measurable cardinals is equivalent to the existenceof a countably additive measure on the reals which measures all sets of reals andextends Lebesgue measure (Ulam [3]). However, the existence of real-valued mea-surable cardinals, and particularly if the continuum is real-valued measurable, hasan array of Set Theoretic consequences reaching beyond measure theory. For ex-

ample: a real-valued measurable cardinal has the tree property (Silver [4]); if thereis a real-valued measurable cardinal, then there is no rapid p-point ultrafilter on N(Kunen); the dominating invariant d cannot equal a real-valued measurable cardinal(Fremlin). And further, if the continuum is real-valued measurable then 20 holds(Kunen); and for all cardinals between 0 and the continuum we have 2 = 20

(Prikry [5]); see [2].On the other hand, there are other properties of Solovays model that have not

been shown to follow from the mere existence of real-valued measurable cardinals:for example, the covering invariant for the null ideal cov(N) has to equal the con-tinuum.

Thus, Fremlin asked ([2, P1]) whether every real-valued measurable cardinalcan be obtained by Solovays method (the precise wording is: suppose that isreal-valued measurable; must there be an inner model M V such that is

measurable in M and a random extension M[G] V of M which contains P?).The question was answered in the negative by Gitik and Shelah ([6]). The broaderquestion remains: what properties of Solovays model follow from the particularconstruction, and which properties are inherent in real-valued measurability?

2000 Mathematics Subject Classification. 03E35, 03E55.Key words and phrases. real-valued measurable.The first author is supported by Chubu University grant S55A. The second and third authors

were supported by the United States-Israel Binational Science Foundation (Grant no. 2002323)and NSF grant No. NSF-DMS 0100794. Publication no. 763 in the list of Shelahs publications.

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2 SAKAE FUCHINO, NOAM GREENBERG, AND SAHARON SHELAH

In this paper we present a new construction of a real-valued measurable cardinaland identify a combinatorial, measure-theoretic property that differentiates between

Solovays model and the new one.The property is the existence of what we call general sequences - Definition 4.5.

A general sequence is a sequence which is sufficiently random as to escape all sets ofmeasure zero. Standard definitions of randomness are always restricted, in the sensethat the randomness has to be measured with respect to a specified collection ofnull sets (from effective Martin-Lof tests to all sets of measure zero in some groundmodel). Of course, we cannot simply remove all restrictions, as no real escapes allnull sets. However, we are interested in a notion that does not restrict to a specialcollection of null sets but considers them all. One way to do this is to change thenature of the random object - here, from a real to a long sequence of reals, and tochange the nature of escaping. We remark here that the following definition echoes(in spirit) the characterization of (effective) Martin-Lof randomness as a string,each of whose initial segments have high Kolmogorov complexity.

We thus introduce a notion of forcing Q. We show that if is measurable(and 2 = +), then in VQ , (which is the continuum) is real-valued measurable(Theorem 3.18). We then show that in Solovays model, the generic (random)sequence is general (Theorem 4.6); and that in the new model, no sequence isgeneral (Theorem 4.14).

1.1. Notation. PX is the power set of X. A B is set difference. denotesinclusion, not necessarily proper; denotes proper inclusion.

The reals R are identified with Cantor space 2. If 2


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MODELS OF REAL-VALUED MEASURABILITY 3

P Q denotes the fact that P is a complete suborder ofQ. IfP Q and G isthe (name for the) P-generic filter, then Q/G is the (name for the) quotient ofQ

by G: the collection of all q Q which are compatible with all p G.IfP Q, a strong way of getting P Q is having a restriction map q q P

from Q to P: a map which is order preserving (but does not necessarily preserve), and such that for all q Q, q P P q Q/G. IfB is a complete subalgebraof a complete Boolean algebra D then there is a restriction map from D to B;

d B =B

{b B : b d} is in fact the largest b B which forces that d D/G;D/G = {d D : d B G}.

IfB is a complete subalgebra of a complete Boolean algebra D then we let D : Gbe the (name for the) quotient ofD by the filter generated by the generic ultrafilterG B; D : G is the completion of the partial ordering D/G.

1.2. Measure theory.

Notation; recollection of basic notions. Recall that a measurable space is a set Xtogether with a measure algebra on X: a countably complete Boolean subalgebra ofPX, that is some S PX containing 0 and X and closed under complementationand unions (and intersections) of countable subsets of S. A probability measure ona measure space (X, S) is a function : S [0, 1] which is monotone and countablyadditive: (0) = 0, (X) = 1 and whenever {Bn : n < } S is a collection ofpairwise disjoint sets, then (Bn) =

(Bn). All measures we encounter in this

work are probability measures.Let be a measure on a measurable space (X, S). Then a -null set is a set

A S such that (A) = 0. We let I be the collection of -null sets; I is acountably complete ideal of the Boolean algebra S; we can thus let B = S/I;

this is a complete Boolean algebra and satisfies the countable chain condition. ForA S, we let [A] = A + I B. We often confuse A and [A], though. Welet , = etc. be the pullback of the Boolean notions in B. Namely: A B if[A] B [B] (iff A B I), etc. We also think of as measuring the algebraB; we let ([A]) = (A).

Definition 1.1. Let S R be two measure algebras on a space X, and let bea measure on S and be a measure on R. We say that is absolutely continuouswith respect to (and write ) if I I; that is, if for all A S, if (A) = 0then (A) = 0.

(Of course, if , A S and (A) = 1 then (A) = 1).If then the identity S R induces a map i : B B which is a complete

Boolean homomorphism. If I = I S then i is injective.

Definition 1.2. Let be a measure on (X, S), and let A S be a -positiveset. We let A, the localization of to A, be restricted to A, recalibratedto be a probability measure: it is the measure on (X, S) defined by (A)(B) =(B A)/(A).

If A = A then A = A so we may write a for a B. We have a

and Ba = B( a); under this identification, the natural map i : B Ba isgiven by i(b) = b a. If a = 1 (so = a) then i is not injective.


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Products of measures. If for i < 2, i is a measure on a measurable space (Xi, Si),then there is a unique measure 0 1 = 0 1 defined on the measure algebra

on X0 X1 generated by the cylinders, i.e. the sets A0 A1 for Ai Si, suchthat (0 1)(A0 A1) = (A0)1(A1) for all cylinders A0 A1. We recall Fubinistheorem: For any measurable A X0 X1, we have

(0 1)(A) =

X0

1(Ax) d0(x),

where for x X0, Ax = {y X1 : (x, y) A} is the x-section of A.We note that localization commutes with finite products:

(0B0) (1B1) = (0 1)(B0 B1).

We can generalize the notion of absolute continuity.

Definition 1.3 (Generalized absolute continuity). Suppose that measures (X, S)

and measures (Y, R), and further that there is a Boolean homomorphism i : S R. We say that if whenever A S and (A) = 0 then (i(A)) = 0.

If i is injective then we dont really get anything new (we may identify S withits image). In any case, the map i induces a Boolean homomorphism from B toB.

The standard example is of course if S = S0 and R is the algebra generated byS0 S1 as above. We then let i(A) = A X1 and get 0 1 0. The map i isinjective and induces a complete embedding

i010 : B0 B0 1 .

The following is an important simplification in notation.

Notation 1.4. Unless otherwise stated, we identify B0 with its image under i010 .

Thus A S0 is identified with A X1.

Thus if Ai Si then A0 A1 = A0 A1.

The restriction map from B01 onto B0 is nicely defined: for measurable A X0 X1, we let

A 0 = {x X0 : 1(Ax) > 0};

this is the measure-theoretic projection of A onto X0. If A =01 A then A

0 =0 A 0, so we indeed get a map from B01 onto B0 , and [A]01 B0 =

[A 0]0 .We make use of the following.

Lemma 1.5. Let be a measure on X and for i < 2 let i be a measure on Yi.

LetBi X Yi and let Ai = Bi . Then A0 A1 = 0 iff B0 B1 =01 0.Proof. To avoid confusion, in this proof we dont use the convention 1.4.

Suppose that A0 and A1 are -disjoint. Then A0 Y0Y1 A1 Y0Y1 =01 0.Also, Bi i Ai Yi so Bi Y1i 01 Ai Y0 Y1; it follows that B0 Y1and B1 Y0 are 01-disjoint.

Suppose that B0Y1 and B1Y0 are 01-disjoint. Consider (B0Y1) 1; AsB1Y0 is a cylinder in the product (XY1)Y0, we have (B0Y1) 1B1 =1 0.However, (B0 Y1) 1 = A0 Y1. Now reducing from X Y1 to X we getA1 = B1 is -disjoint from A0.


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Infinite products. Iterating the two-step product, we can consider products of finitelymany measures. However, we need the more intricate notion of a product of infin-

itely many measures. Countable products behave much as finite products do. Let,for n < , n be a measure on a measurable space (Xn, Sn). Again, a cylinder isa set of the form

n 0, but (

An) = 0;

in this case we can use the measure

(nAn), but

An cannot be defined.To better understand uncountable products, we notice that a countable product

can be viewed as a direct limit of finite products. Namely, we let a finite cylinder bea set of the form

n


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Definition 1.6. Let u . A pure local product measure on u is a measure onSu of the form u(m{}B) for B S{}. A local product measure on u is ameasure on Su of the form B, where is a pure local product measure on u.

We will mention other measures (such as a measure witnessing that a cardinalis real-valued measurable); but when it is clear from context that we only mentionlocal product measures, we drop the long name and just refer to measures andpure measures.

If is a local product measure on u then we let u = u and call u the support(or domain) of .

Topology. We note that every Ru is also a topological space (which can be viewedas the Tychonoff product ofR{} for u). However, when u is uncountable, thenthe Borel subsets ofRu properly extend Su. This is not a concern of ours becausethe completion of any local product measure measures the Borel subsets ofRu. We

thus abuse terminology and when we say Borel we mean a set in Su; so for us,every Borel set has countable support. [In some texts, sets in Su are called Bairesets. We choose not to use this terminology to avoid confusion between measureand category.]

Recall that a measure which is defined on the Borel subsets of a topologicalspace is regular if for all Borel A, (A) is both the infimum of(G) for open G Aand the supremum of (K) for compact K A. [Thus up to -measure 0, eachBorel set is the same as a 02 (an F) set and as a

02 (a G) set.] Lebesgue measure

is regular, and a localization of a regular measure is also regular. Also, regularity ispreserved under products; again note that even with uncountable products, everymeasurable set has countable support and so the closed sets produced by regularityhave countable support.

Corollary 1.7. Every local product measure is regular.Random reals. Let be a local product measure. Forcing with B is the same asforcing with I+ = Su I, ordered by inclusion. The regularity of shows thatthe closed sets are dense in I+ . It follows that a generic G I

+ is determined by

{rG} = BGBV[G].

We have B G iff rG BV[G]. We have rG

{AV[G] : A V is co-null}; andconversely, ifW is an extension ofV and r W lies in

{AV[G] : A V is co-null},

then G = {A I+ : r AW} is generic over V and r = rG.

Suppose that , are local product measures and that u u = 0. Then is a local product measure. Recall that we have a complete embedding i : B B . Thus if G B is generic then G = (i )

1G is generic for B. In fact,

rG = rG u.

Quotients are measure algebras. Let V[G] be any generic extension of V. There is

a canonical extension of to a measure on SV[G]u , which we denote by V[G]. For

if = (

u(m{}B))B then we can let V[G] = (

u(m{}B

V[G] ))BV[G].

The usual absoluteness arguments show that indeed V[G] is an extension of , anddoes not depend on the presentation of .

Again let and be local product measures on disjoint u = u , v = u . Wemake use of the following.


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Fact 1.8. Let G B be generic. Then the map A AV[G]rG induces an isomor-phism from B : G to BV [G] .

In particular, B B : G is a measure algebra.

Proof. Let , : B B : G be the quotient map. We know that

1,(B : G {0}) = B/G

(the partial ordering). Thus for A Suv, we have

,([A]) > 0 [A ] G

rG (A )V[G] V[G](AV[G]rG) > 0,

The last equivalence follows from the fact that (A )V[G] = AV[G] V[G]; againwe use absoluteness. Thus we may define an embedding , : B : G BV[G]

by letting ,(,([A])) = AV[G]rGV[G] . It is clear that , preserves the

Boolean operations., is onto: every set in the random extension is determined by a set in the

plane in the ground model (see [7, 3.1]). For any countable v v, every B-namey for an element of Rv

corresponds to a 03 function fy : Ru Rv

defined byfy(x)(i)(n) = k x [[y(i)(n) = k]]B (fy can be taken to be

03 because

[[y(i)(n) = k]] can be taken to be either a 02 or a 02 set.) The function fy has the

property that fV[G]y (rG) = yG. Let C be a B-name for a Borel subset ofR

v. Thealgebra B is c.c.c., so in V, there is some countable v

v and some B-name C

for a Borel subset ofRv

such that B C = C Rvv

. We can let

A = {(x, fy(x)) : x [[y C]]B} R

vv

where fy ranges over 03 functions from R

u

to Rv

; thus A is Borel and (AG

)rG =CG. [However, in the sequel, we do not use the fact that , is onto.]

Commuting diagrams. We thus have the following diagram:

B,

// B : G,

// BV [G] .

Suppose now that is a local product measure on u; , are local productmeasures on v0, v1, and u, v0, v1 are pairwise disjoint. Let =

. Let G B

be generic. For the rest of the section, we retract our convention 1.4. We thushave a complete embedding i : B B . This embedding induces a completeembedding : B : G B : G.

Lemma 1.9. The following diagram commutes.

B,

//

i

B : G

,// BV [G]

iV[G]

V[G]

B

,// B : G

,// BV[G]


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Proof. Let A Suv0 , and let

a = ,

([A]

);a = (a) = , ([A R

v1 ]) ;

b = ,(a) =

AV[G]rGV[G]

; and

b = ,(a) =

(A Rv1)V[G]rG

V[G]

.

The desired equation iV[G]

V[G](b) = b follows from the fact that

AV[G]rG Rv1V[G] = (A Rv1)V[G]rG .

Note that iV[G]

V[G]is measure-preserving.

Next, suppose that , are local product measures on u0, u1 and that is a local

product measure on v; and that u0, u1, v are pairwise disjoint. We let = .As i is a complete embedding, we know that if G B is generic, then G =

(i)1G is also generic. The map i induces a complete embedding

from

B : G to B : G .Also, as V[G] V[G ] we have (relying on absoluteness) a measure-preserving

embedding ,, : BV [G ] BV [G ] , given by [B]V [G ]

BV[G ]V[G ]

.

Lemma 1.10. The following diagram commutes:

B,

//

i

B : G

,// BV[G ]

,,

B

,// B : G ,

// BV [G ]

Proof. Let A Su0v. We let:

a = , ([A]) ;

a = (a) = ,

[A Ru1 ]

;

b = ,(a) =AV[G ]rG

V [G ]

; and

b = ,(a) = (A R

u1)V[G ]

rG V [G ] .We want to show that ,, (b) = b

. Letting B = AV[G ]rG and B = (A Ru1)V[G ]rG ,

we show that B = BV[G ]. We know, though, that rG = rGrG , from whichwe deduce that B = AV[G]rG . The conclusion follows from absoluteness.

In our third scenario, we have , which are local product measures on disjointv, u; and we let = B be some localization of . In this case we have a projectioni : B B . Let G B be generic; then G = (i

)

1G is generic, but in factcontains no less information; so we denote the extension by V[G].


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Lemma 1.11. The following diagram commutes:

B

,//

i

B : G

,

((QQQQ

QQQQ

QQQQ

Q

BV[G]

B,

// B : G

,

66mmmmmmmmmmmmm

Proof. As usual we take A Svu and follow [A] along the diagram. Wehave ,, (, ([A] )) = ,([A]); and , (, ([A] )) =

AV[G]rG

V[G]

and

,(,([A])) = AV[G]

rG V[G] ; the latter two are equal because rG = rG .

Our last case is perhaps the easiest (in fact we do not use it later but we include itfor completeness.) Suppose that u and v are disjoint and that , are local productmeasures on u, v respectively. Suppose that C B is positive; let = C. LetG B be generic. Then by absoluteness V[G] = V[G]CV[G]. Note that unlike

the previous cases, the Boolean homomorphism iV[G]

V[G]is not measure-preserving.

Lemma 1.12. The following diagram commutes.

B,

//

i

B : G

,// BV [G]

iV[G]

V[G]

B

,// B : G

,// BV[G]

Proof. Immediate, because for A Suv, i([A]) = [A] (which is the same as[A (Ru C)]).

2. Solovays construction

We hope that the gentle reader will not be offended if we repeat a proof ofSolovays original construction of a real-valued measurable cardinal, starting froma measurable cardinal. The exposition which we give is different from the one foundin most textbooks, indeed from the one given by Solovay in his paper; since in therest of this paper we shall elaborate on this proof, we thought such an expositionmay be useful.

Let be a measurable cardinal; let j : V M be an elementary embedding ofV into a transitive class model M with critical point , such that M M.

We move swiftly between M, V, M[G] and V[G]. Whenever necessary we indi-cate where we work, but many notions are absolute and there is not much dangerof confusion.


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The forcing Solovay uses is P = Bm , i.e. forcing with Borel subsets of R of

positive Lebesgue measure. We show that after forcing with P, is real-valued

measurable.We have j(P) =

Bmj()

M= Bmj() . Also, P M and P = (Bm)

M.

Let G P be generic over V. Then G is generic over M. We have the followingdiagram:

j(P)m,m[,j())

// j(P) : Gm,m[,j())

//Bm[,j())

M[G].

For shorthand, we let = m,m[,j()) and we let be the pullback to j(P) : G of

m[,j())M[G] by m,m[,j()) .

Let A be a P-name for a subset of . In M, j(A) is a j(P)-name for a subset

of j(). Let bA =

[[ j(A)]]j(P)

M(note A bA is in V) and in V[G] let

(A) = ((bA)). We now work in V so we refer to the objects defined as names.

Lemma 2.1. Suppose that a P, that A, B are P-names for subsets of , and thata P A B. Then a P (A) (B).

Proof. The point is that j(a) = a. Let G P be generic over V such that a G.In M, a j(P) j(A) j(B). Let b = bA a. As a G we have (a) = 1j(P) : G so(b) = (bA). However, in M, b j(P) j(B) (as it forces that j(A) j(B) andthat j(A)) and so b bB. It follows that (bA) (bB) so (A) (B). AsG was arbitrary, a P (A) (B).

It follows that , rather than being defined on names for subsets of , can bewell-defined on subsets of in V[G]. The following lemmas ensure that is indeeda (non-trivial) -complete measure.

Lemma 2.2. Let A be in V and let G P be generic over V. If j(A)then (A) = 1 and if / j(A) then (A) = 0.

Proof. Suppose that j(A). Then in M, 1j(P) j(A). Thus bA = 1j(P) so(bA) = 1j(P) : G. Thus (A) = 1. On the other hand, if / j(A) then in M, nob j(P) forces that j(A), so bA = 0; it follows that (A) = 0.

Lemma 2.3. Suppose that Ann


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Lemma 2.4. Suppose that < and that A


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for all w, (m{}B)(B

) < (or the other way round). Let A =

w B

.

Then (A) = 1 but (A) = 0.

Let w = { v : B =m{} B

}. We assume that w = 0 for otherwise

were done. Let A =

w B (and similarly define A

). It is sufficient to showthat (A), (A) > 0; it will then follow that 0 is a pure extension of A,and similarly for 0. Suppose that (A

) = 0. Let w; let a = m{}(B

B), c = m{}(B B

) and b = m{}(B

B

). The assumption is that

wb+c

a+b+c= 0. However, for each w, b

a+b b+c

a+b+c, which means

that

w(m{}B)(B

) = 0, so (A

) = 0 (and of course (A) = 1).

For , Q, if u u = 0 then , and so and are compatible. Thefollowing is the generalization we need:

Lemma 3.5. Let u be a set of ordinals and let , Qu. Then Qu iff exp .

Proof. Suppose that exp . We may assume that v = uu = 0. By lemma 3.4,find some pure on v, some pure 1, 1 and some C

, C such that = ( 1)C,

= ( 1)C . Let = 1 1. We have (C C) > 0 for otherwise, by lemma1.5, C , C are -disjoint and would witness that exp . Then (CC)is a common extension of and ; for example, (C C) = ( 1)C .

Remark 3.6. If then there is some on u u which is a common extensionof and . In fact, the common extension constructed in the proof of lemma 3.5is the greatest common extension of and in Q (thus this extension does notdepend on the choice of ).

3.1.2. Characterization of the generic. Let u be a set of ordinals, and let G Qube generic over V. Let

AG = {BV[G] : for some G, (B) = 1}.Lemma 3.7. AG is not empty.

Proof. Let FG = {BV[G] : B is closed and for some G, (B) = 1}, and letBG = FG. We show that BG = AG and that BG is not empty.

For the first assertion, recall (corollary 1.7) that every Q is a regular measure.Let Qu and let B be of -measure 1. There is some closed A B of positivemeasure, so A Qu. Thus by genericity, for every B such that (B) = 1 forsome G, there is some closed A B and some G such that (A) = 1. Thisshows that BG = AG.

Next, we note that FG has the finite intersection property. Let F FG be finite.For B F let B G witness B FG. There is some G which extends all Bfor B F. Then (F) = 1 which implies that F = 0. As RuV[G] is compact,BG = 0.

In fact,

Lemma 3.8. AG is a singleton {sG}.

Proof. Let < and let n < . There is some G and some 2n such that([]) = 1. For given any we can extend it to some such that u and thenextend locally to some such that ([]) = 1 for some 2n.

As usual,


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Lemma 3.9. V[G] = V[sG].

Proof. In fact, G can be recovered from sG because for all Qu, G iff for allB such that (B) = 1 we have sG BV[G]. For if / G then there is some Gsuch that . By lemma 3.5, there is some B such that (B) = 0 and (B) = 1.Then sG / BV[G].

3.1.3. The size of the continuum. Here is an immediate application:

Lemma 3.10. Qu adds at least |u| reals.

Proof. Let G be generic and let sG be the generic sequence. We want to show thatfor distinct , u we have sG = s

G . Let G be such that , u

.

m{,} so m{,}. Let A = {(x, y) R{} R{} : x = y} be the complement

of the diagonal. Then m{,}(A) = 1 so (A) = 1. Thus sG AV[G]. But

AV[G] = {(x, y) R{}V[G]

R{}V[G]

: x = y}. Thus sG = sG .

3.2. More on local and pure extensions. Let Q. The collection of localextensions of (ordered by ) is isomorphic to B, so we identify the two.

Lemma 3.11. Let Qu. ThenB Qu( ).

Proof. Let A, B B. Then A and B are compatible in B iff (A B) > 0 iffA, B are compatible in Q.

Let Ann 0. Then An is acommon extension of and An.

It follows that sG is a string of random reals.

Remark 3.12. For all u v we have Qu Qv; we do not need this fact.

Definition 3.13. Let Qu and let U Qu. We say that determines U ifU B is dense in B.

We say that Qu determines a formula of the forcing language for Qu if determines { Qu : decides }. Of course, this depends on u, so if not clearfrom context we will say u-determines. Informally, determining means that is transformed to be a statement in the random forcing B, which is a simple notion,compared to formulas ofQu. If determines pertinent facts about a Qu-name thenthat name essentially becomes a B-name.

For a formula of the forcing language for Qu and Qu we let

[[]]u =

B{b B : b Qu }.

Then u-determines iff [[]]u [[]]

u = 1B . Recall that if then

so there is a natural map i : B B (which is a measure-preserving embeddingif is a pure extension of ). For all a B, if i(a) = 0 then i

(a) Q a,

so for all , [[]]u B i([[]]

u). Thus if determines then so does and in this

case [[]]u = i([[]]

u). If also pur then these Boolean values have the same

measure: ([[]]u) = ([[]]

u).

We now prove that determining a formula is prevalent. Here and in the rest ofthe paper we often make use of sequences of pure extensions. This gives us some


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closedness that the forcing as a whole does not have; the situation is similar to thatof Prikry forcing. We should think of pure extensions as mild ones.

A pure sequence is a sequence ii


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Let DG = { : G & < }. This is a directed system (under Q).Note that from DG we can recover and . We thus let, for = DG,

= , , : B BV [G ] be the quotient by G (this of course depends onG and not on alone, but we suppress its mention). Lemmas 1.9, 1.10 and 1.11and the discussion between them show that for any = = in DG

and any a B we have V[G ] ((a)) =

V[G ] ((i

(a))).

Let be a formula of the forcing language for Q. For = DG we let

() = V[G ] (([[]]

)). The analysis above shows that if

are in DG then() (), and that if -determines then () = () for all . Wetherefore let G() = supDG (). To calculate G() it is sufficient to take thesupremum of () over a final segment of DG (or in fact any cofinal subset ofDG). If some DG determines then () is eventually constant and we getG() = maxDG () which equals () for any which determines .

Remark 3.17. This is important. Suppose that M is an inner model of V. Thenwe can work with this scenario mostly in M: well have all the ingredients inM (so , are inaccessible in M, Q,Q are in the sense of M) but the sequence will not be in M. Thus if G QM is generic over V then the entire system(DG, , (), . . . ) will be in V[G] but not in M[G] (of course G is generic overM too). We can still make, in V[G], the above calculations of G() for M

(although determining and the calculation of [[]] and () for each particular

will be done in M or M[G]).

3.3. Real-valued measurability. In this section we prove the following:

Theorem 3.18. Suppose that there is an elementary j : V M with critical point such that M


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Lemma 3.19. Let P and A, B N. Suppose that P A B. ThenP fG(A) fG(B).

Proof. As we had in our discussion of Solovays construction, j P is the identity.So in M, j(P) j(A) j(B). Let G P be generic and suppose that G. Forany = DG such that we have so in M, j(P) j(A) j(B)

so [[ j(A)]] B [[ j(B)]] so ( j(A)) ( j(B)). As this is true for

a final segment of DG we have G( j(A)) G( j(B)). [Note that in thisproof we didnt need any particular U.]

It follows that fG induces a function on subsets of in V[G] (rather than onlyon their names). We show this function is the desired measure on .

Lemma 3.20. LetA be in V. If j(A) thenP fG(A) = 1 and if / j(A)thenP fG(A) = 0.

Proof. Suppose that j(A). Then in M, every condition in j(P) forces thisfact. Let G P be generic. It follows that for all DG, [[ j(A)]]

= 1B so

G( j(A)) = 1.We get a similar argument if / j(A).

Lemma 3.21. Let Bnn


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Now there are |N|


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4.1. General sequences in Solovays model.

Theorem 4.6. Let be inaccessible. Then in VBm , the random sequence is -general for all regular, uncountable < .

This relies on the following well-known fact:

Fact 4.7. Let P be a notion of forcing which has the -Knaster condition for all reg-

ular uncountable < , and let A V. Then (in VP), ([A]


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and so r w / B. The noncountable club C = ([W]


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forces that w is contained in w, the collection of all such as which appeared inthe construction ( is regular, so < and so w has size < ).

In fact, for every w [A] . We can present U as a disjoint unionof cylinders Un; for some n we must have (Un B)/(Un) > . Then Un is a

pure measure and is as required.

We need a certain degree of uniformity.

Lemma 4.12. LetB R2 be a positive Borel set. Then there is some positive A R such that for all < 1 there is some positive C R such thatm2(AC)(B) > .

Proof. Let X V 2 be countable such that B X. Let C0 be the measure-theoreticprojection of B onto the y-axis (of course C0 X). C0 is positive, so we can picksome r C0 which is random over X. Let A = B

r = {x R : (x, r) B} bethe section defined by r; since r C0, A is positive. Note that in X there is aname for Br

, where r is a name for the generic random real.Let > 0 be in X. By regularity of Lebesgue measure, there is some clopen set

U R such that m(UA) < .3 Of course U X. Then there is some positiveC C0 in X such that CBm m(UB

r) < .For almost all r C (those that are random over X), we have m(UBr) < .

For such r, m(A Br) m(A U) + m(U Br) 2. So by Fubinis theorem,m2(A C B) =

C

m(A Br) d r 2m(C); we get that m2(A C)(B) 2/m(A).4

Corollary 4.13. Let, Q and let be pure; assume u u = 0. LetB B .Then there is a localization of such that for all < 1 there is some pure

which is a localization of and such that (B) > .

Proof. What we need to note is that the proof of the previous lemma holds for (in place ofm m) (we just use the relevant measure algebra); we get a set A B

such that for all > 0 there is some C B such that (A C)(B) > 1 .

2Yes, we mean X H(). Complaints are to be lodged with set models of ZFC.3Let V A be open such that m(VA) < /2; and recall that every open set is an increasing

union of clopen sets.4We glossed over uses of the forcing theorem over X, which is not transitive. We really

work with Xs collapse and use absoluteness. For example, we got C X such that C Bmm(UBr

) < . Let : X M be Xs transitive collapse. Then in M, (C) forces (in BMm )

that m((U)(B)r

) < . If r C is random over M, then in M[r], m((U)(B)r ) < . But

(B)M[G] = B M[G] and similarly for U. Thus indeed m(UBr) < as we claimed.


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Fix some > 0. Get the appropriate C; we have

(A C B)

(A C) < .

By Lemma 4.11, we can find some cylinder C B sufficiently close to C so thatboth C(C) > 1 and (C)/(C) < 1 + ; from the first we get

(C C)

(C)< .

Note that A C B A (C C) (A C B). Combining everything, we get

(A C B)

(A C)

(A (C C)) + (A C B)

(A C)=

(A)(C C)

(A)(

C)

+ (A C B)

(A C)

(A)(C)

(A)(

C)

+ (1 + ).

We can thus let = A and = C; the latter is pure because C is a cylinder.We get (B) 1 2 2 which we can make sufficiently close to 1.

4.3. In the new model.

Theorem 4.14. Suppose that is Mahlo for inaccessible cardinals, and that 2is at most the least inaccessible (and is regular). Then in VQ , there are no -general sequences.

In fact, we prove something stronger:

Theorem 4.15. Suppose that is Mahlo for inaccessible cardinals, and that 2is regular, and is at most the least inaccessible. Then there is a -null sequence B

of length such that in VQ

, no -sequence of reals r escapes

B.(We have here identified B as it is interpreted in V and in VQ . Of course,

for every -null B, if B = i


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stationary S1 S0, the function is constant. Next, we find S2 S1 suchthat on S2:

u and otp u are constant; Under the identification of one Ru to the other by the order-preserving

map, (u ) is constant; Under the identification of one Ru to the other by the order-preserving

map, B is constant; k() is a constant k.

By these constants, and using Corollary 4.13, we can find some , a localizationof for S2, such that for all < 1 and all S2, there is some pure whichis a localization of , such that (

)(B) > .

We now amalgamate countably many s in the following way. Pick an increas-ing sequence nn qn, where qn is a sequence of

rational numbers in (0, 1) chosen so that

n


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The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J.

Safra Campus, Givat Ram, Jerusalem 91904, Israel

Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State Univer-sity of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA

Sakae Fuchino, Noam Greenberg and Saharon Shelah- Models of Real-Valued Measurability

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Transcript of Sakae Fuchino, Noam Greenberg and Saharon Shelah- Models of Real-Valued Measurability