Weak Measure Extension Axioms

Joan E Hart

Union College Schenectady NY USA

and

Kenneth Kunen

University of Wisconsin Madison WI USA

hartjunionedu and kunenmathwiscedu

February

Abstract

We consider axioms asserting that Leb esgue measure on the real

line may b e extended to measure a few new nonmeasurable sets

Strong versions of such axioms such as realvalued measurabilityin

volve large cardinals but weak versions do not We discuss weak

versions which are sucienttoprovevarious combinatorial results

such as the nonexistence of Ramsey ultralters the existence of ccc

spaces whose pro duct is not ccc and the existence of S and L spaces

We also prove an absoluteness theorem stating that assuming our ax

iom every sentence of an appropriate logical form which is forced to

b e true in the random real extension of the universe is in fact already

true

Intro duction

In this pap er a measure on a set X is a countably additive measure

whose domain the measurable sets is some algebra of subsets of X

The authors were supp orted by NSF Grant CCR The rst author is grateful

for supp ort from the University of Wisconsin during the preparation of this pap er Both

authors are grateful to the referee for a numb er of useful comments

INTRODUCTION

We are primarily interested in nite measures although most of our results

extend to nite measures in the obvious way By the Axiom of Choice

whichwealways assume there are subsets of which are not Leb esgue

measurable In an attempt to measure them it is reasonable to p ostulate

measure extension axioms of the following form

Denition If is any cardinal and is a measure on the set X then

ME X holds i whenever we choose a family E of or fewer subsets of

X thereisameasure on X which extends such that each set in E is

measurable ME denotes ME where is Lebesgue measureon

For ME X holds for every nite measure but for innite

it can dep end on X and the underlying mo del of set theory Regardless

of the set theory there is always some separable atomless probabilityspace

X suchthatME X is false by Theorem of Grzegorek see

also x In this pap er we are concerned mainly with ME not arbitrary

ME X but in applications of ME it is often convenient to replace

by or by with the usual pro duct measure This is justied bythe

following

Prop osition Let be a nite Borel measure on the compact metric

space X ThenME implies ME X Furthermore ME is equivalent

to ME X unless is a countable sum of point masses

X from ME let f X b e a Borel Pro of To derive ME

measurable function such that is the induced measure c f where

c X Then we can extend to measure a family E of subsets of X by

extending to measure ff E E EgConverselyif is not a sum of

p oint masses we can x a closed K X of p ositive measure suchthat

restricted to K is atomless We can then derive ME from ME X using

a function g K such that c g where c K

Note that if isacountable sum of p oint masses then ME X isa

triviality since then every subset of X is measurable

Now consider ME for various innite ME is false under CH or MA

for numerous reasons see b elow Nevertheless ME and also ME are

INTRODUCTION

consistent with c the continuum b eing In general for uncountable





one can get ME together with c

Theorem Carlson Assume that in the ground model V is some



innite cardinal with Let V G be formedfrom V by adding or

morerandom reals Then ME holds in V G

In particular if CH holds in V and thenwegetamodelof

ME plus c More generally for regular we can get mo dels



 

of ME with either c or c Furthermore for small eg b elow

the rst weakly Mahlo cardinal ME implies MA for the partial order



which adds one random real see Corollary and hence c Carl

sons pap er discusses applications of ME forvarious uncountable

cardinals where is the usual pro duct measure to normal Mo ore space

problems whereas our pap er concentrates on applications of ME that is

ME

The emphasis of this pap er is on small butwe remark briey on ME

for larger values which leads naturally to large cardinal axioms By the

metho d of Solovay see also the assumption of ME plus c is

equiconsistentwitha weakly compact cardinal By Ulam the existence

ofarealvalued measurable cardinal is equivalent to what one might call

ME that is Leb esgue measure can b e extended to measure all sets of

reals simultaneouslySoby SolovayME is equiconsistentwiththe

existence of a twovalued measurable cardinal For a discussion of PMEA

whichinvolves extending measures on various see Fleissner

Weturnnow to applications of ME These are all statements which hold

in random real extensions and would thus would b e easy to prove from a real

valued measurable cardinal using Solovays Bo olean ultrap ower metho d

but require some care to derive from the weaker ME Inx we establish an

absoluteness theorem whichsays that assuming ME if a statement of a

certain simple logical form is true ab out in random real extensions then

is already true in V The form of enables us to pro duce in V ob jects

which can b e constructed from a single random real Some applications are

given by the following theorem

INTRODUCTION

Theorem ME implies

thereare ccc topological spaces X and Y such that X Y is not ccc

therearestrong S and Lspaces

thereisanuncountable entangled set

As usual ccc denotes the countable chain condition By Galvin CH

implies that ccc is not pro ductive whereas MA CH implies that ccc is

pro ductive see eg Theorem of As is well known see eg

Exercise C of pro ductivityof ccc is the same whether we deal with

top ological spaces or with partial orders and work on pro ductivityof ccc

usually deals with the partial orders directly

Roitman showed that in random real extensions of V ccc partial or

ders P and Qwith P Q not cccmay b e constructed from a single random

real In the same pap er she constructed strong S and Lspaces from a sin

gle random real The fact that uncountable entangled sets are added bya

random real is due to To dor cevic see page of for a pro of It follows

almost immediately from these facts and Solovays Bo olean ultrap ower con

struction that the conclusions to Theorem follow from a realvalued

measurable cardinal see also Fremlin for a discussion To apply our abso

luteness result to pro duce these results from the weaker assumption of ME

however we exploit the form of the construction of the desired ob jects from

the random real this is discussed in x Actually Theorem follows

directly from Theorem byTodorcevic

While our absoluteness result applies to ob jects which can b e constructed

from one random real some further applications of ME do not seem to t

this pattern For example ME implies MA for the partial order which

adds one random real Corollary which in turn has a number of well

known consequences eg every subset of of size is of rst category

Of course this refutes CH Also Corollary ME implies that no

Leb esgue measurable set of p ositive measure can b e an increasing union of

Leb esgue nullsets This also refutes CHaswell as full MA Finallywe

mention

Theorem ME implies that therearenoR amsey ultralters

INTRODUCTION

A Ramsey or selective ultralter is a nonprincipal ultralter U on



such that every partition has a homogeneous set in U Asiswell

known this implies that for each nite n U is also Ramsey for partitions

on ntuples Under CH or MA there is a Ramsey ultralter as is easily seen

by W Rudins construction of a P p oint ultralter It is already known

that there are no Ramsey ultralters in the mo del obtained by adding at



least c random reals Using this metho d of pro of Fremlin shows that a

realvalued measurable cardinal refutes the existence of Ramsey ultralters

A pro of of Theorem may b e patterned after the argument in but we

give a dierent argument which also improves the result of to

Theorem Let V G be formedfrom V by adding or morerandom



reals Then in V G therearenoRamsey ultralters

Although the metho d of alone do es not seem to prove this in

proving Theorems and weemulate to refute a prop ertyweaker

than Ramsey known as rapid Pp ointor semiselectiv e



A sp ecial case of Theorem is that ME b ecomes true if weaddc

random reals But while adding random reals suces for Theorem



adding random reals may not b e enough to get ME To see this ob



serve that ME is false if the wellorder on the cardinal c is in the algebra

generated by rectangles since byFubinis Theorem the sides of the rect

angles will form a countable collection of subsets of c equivalentlyof

which cannot b e measured byany atomless additive probability measure

In particular by or ME is false under CH or MAFurthermore

supp ose the ground mo del V satises MA CH Then adding random



reals do es not change c soitisstilltrueinV G that the wellorder on c is

in the algebra generated by rectangles so ME is false in V G Further

use of rectangles to derive theorems from ME o ccurs in x

We also cannot replace the by in Theorem if the ground mo del



satises CHthenCH will remain true after adding random reals so

there will b e a Ramsey ultralter in the extension

Theorems and b oth have the same conclusion no Ramsey ul

tralters The pro ofs given in x are similar to o and utilize the same

probabilistic argument although the pro of of Theorem adds a forcing in

gredient Our metho d for refuting Ramsey ultralters in xmay seem a bit

PRESERVING SUPREMA

articial since the argument do es not deal directly with the Ramsey prop

erty at all but rather with a rather technical consequence thereof In x

we present a more natural argument using random graph theory Actually

the metho d in x requires more work in verifying the details than do es the

metho d of x but it derives the lack of Ramsey ultralters directly from a

lemma ab out random graphs on nite sets whichmightbeofsomeinterest

in its own right

Preserving Suprema

Akey ingredientof x s pro of of our absoluteness result is the existence of

a measure algebra in which certain suprema are preserved We b egin this

section by reviewing some basic facts ab out measure algebras and then we

lo ok at the suprema preserving strength of ME for various

Denition If isaprobability measureona set X then B is the

measure algebra of the measurable subsets of X modulo the nul lsets If

is a cardinal and is the usual product measureon then we abbreviate

W W

B by B IfE is a family of measurable sets E abbreviates E

E E

Note that the elements of B are equivalence classes E of measur

able sets with D E i D nE is a nullset B is the measure algebra with

which one forces to add random reals This is equivalent to forcing with

the Baire sets of p ositive measure as in When just doing forcing it is

somewhat simpler to use the Baire sets rather than their equivalence classes

but when discussing algebraic prop erties such as suprema it is somewhat

simpler to work with the Bo olean algebra

As is well known B is a complete Bo olean algebra and the following

lemma relates suprema with unions

W S

Lemma If E is a family of measurable sets then E E for



some countable E E



S

table then E may fail to b e measurable If it is measur If E is uncoun

W S

able then E E but this inequalitymay b e strict for example let E

b e a family of singletons from

PRESERVING SUPREMA

Denition If E is a family of measurable sets with jE j then

W S W S

preserves the supremum E i E is measurable and E E in

B

First welookatpreserving suprema and then we lo ok at a notsolarge

cardinal prop erty to handle preserving suprema

Theorem If EPX and jE j then thereisacountable col lection

SPX such that every measureon X which measures ES preserves

W

E

We shall prove this after p ointing out two corollaries Note rst that this

theorem applies when extending a measure with Edom

Corollary ME X X is not a union of nul lsets

For Leb esgue measure this gives us a form of Martins Axiom Sp eci

cally let MA rr represent MA for random real forcing that is MA rr

says that whenever P is a partial order for adding one random real and D

is a family of no more than dense subsets of P then there is a lter G in

P meeting each D D The following wellknown prop osition relates the

preservation of suprema to MA rr

Prop osition MA rr is equivalent to the statement that is not

the union of Lebesgue nul lsets

Thus by Corollary wehave the following

rr Corollary ME MA

ME c Note that we cannot raise the cardinals In particular



by in Corollary ME is consistentwithc soitcannotimply



MA rr Hence in Theorem if jE j we cannot exp ect to get





jS j We can get an S of size aswe explain after proving Theorem



Our pro of employs the following ancientfactproved in Kunen and

Rao ab out

PRESERVING SUPREMA

Theorem Each subset of is in a algebra generatedbycountably

many rectangles of the form B C where B C

Another easy fact we employinproving Theorem reduces suprema to

disjoint suprema

Lemma Suppose that E is measurable for each Let E

W W S

E E E so E E Then E n





Nowwe preserveour suprema

Pro of of Theorem Fix a collection E fE g By Lemma

wemay assume that E is a disjoint collection By Theorem the set

L fh i g b elow the diagonal is in the algebra generated

by a countable collection fB C n g where each B and C is a

n n n n

S

E Dene F E so that F x i subset of Let E

x E Let

S fE g fF B n gfF C n g

n n

S

E E ObservethatL is in the algebra generated by Let L

the F B F C

n n

By way of contradiction assume that measures ES but fails to pre

W W

serve E thatis E E By Lemma x suchthat

W W S

E E Let G fE g Then G is mea

surable and not null but is partitioned into the E for

S

which are null Dene M fE E g Note that

L and hence also M is measurable in the pro duct measure Each

vertical slice M fy x y M g is contained in a countable union

x

of the nullsets E and is hence a nullset whereas each horizontal slice

y

M fx x y M g contains all but countably many of the E for

y

so M G But then M contradicts Fubinis

Theorem

We turn now to preserving suprema for In order to pin down

the for which ME guarantees that we can preserve these suprema we

dene a notsolarge cardinal prop erty

In the following L is always a countable rst order language containing

the symbol For any ordinal a structure A for L is called a structure

PRESERVING SUPREMA

i A has universe and is in terpreted as the usual well order on

Another structure B for L is called an end extension of A i B is a prop er

extension of A and no member of gets a new element that is if and

b B then b implies b if in addition B is an elementary extension

B

it is called an elementary end extensionoreee See for general background

on elementary end extensions We consider here realvalued elementary end

extendible ordinals

Denition Anordinal has the rveee prop erty orisrveeeieach

structure A has an eeeinsomerandom real extension of the universe

The Compactness Theorem implies that is rveee and the end extension

is obtainable without adding random reals We shall see presently Prop osi

tion that a rveee ordinal is in fact a cardinal it is also w eakly Mahlo

andisweakly compact in L Hence the following theorem applies to pre

serving suprema for a reasonablen umb er of smaller

Theorem If EPX and each uncountable jEj is not rveee then

there is a family SPX such that jS j jE j and such that every measure

W

on X which measures ES preserves E

Again by Prop osition wehave the following

Corollary For such that each uncountable is not rveee

ME MA rr



Since MA rr is false and there are mo dels of ME plus c see

c



Theorem we cannot preserve suprema using just ME Similarlyif

c is realvalued measurable then ME is true and MA rr is false so

the assumption on rveee ordinals cannot b e dropp ed from the corollary or

the theorem

W S

Weprove Theorem bycontradiction assuming E E enables

us to use an ultrap ower in some random real extension to build an eee The

following lemma simplies the construction

Lemma Suppose thereare no uncountable rveeeordinals Then

thereisa structure A such that every proper elementary extension B of A

contains a nonstandard integer

PRESERVING SUPREMA

Pro of Let A enco de for eachuncountable ordinal a structure A

witnessing that is not rveee

To prove Theorem we shall use an ultrap ower based on denable sets

For a structure AletD b e the set of subsets of rst order denable in A

A

from elements of and let F b e the set of functions in rst order denable

A

in A from elements of For any ultralter U on D wehave the ultrap ower

A

F U Aversion ofLo s Theorem for denable functions shows that F U is

A A

an elementary extension of A Although the originalLo s Theorem do es not

apply directly here one can provetheversion we need simply by mirroring

the originals pro of which inducts on the complexity of formulas taking care

in the existential case to pro duce a denable function The extension F U

A

will b e prop er i U is nonprincipal Also F U will b e an mo del i for each

A

with f there is some n so that f fng U function f F

A

Pro of of Theorem Fix EPX such that each uncountable jEj

is not rveee and let jE j Let fE g list E and form the

corresp onding disjoint family in the usual way for each set E

S S S

E and dene E so that E E Let E E n

x for each x E FixA as in Lemma and let

S fE gfE gf D D D g

A

W

Supp ose measures SE and supp ose that do es not preserve E so

W W

E E E inB We shall derive a contradiction

W

E G Let G be a B generic lter over V with E G but

Dene U fD D GgByupward closure of GeachE G

so U is nonprincipal and hence F U is a prop er extension of A

A

Finallyweshow that F U is an mo del which will contradict the

A

choice of ASupposef F and f Then is the disjoint union of

A

the sets C f f ng Since the C form a partition of

n n

unity in the ground mo del V some C is in G so some C is in U

n n

Next weshow that rveee ordinals are weakly Mahlo in V andweakly

compact in L This argument uses nothing sp ecial ab out random real forcing

b esides the fact that conalities are preserved

First observe that the KeislerSilver argumentworks also for forcing extensions

PRESERVING SUPREMA

Lemma Let be any uncountable regular cardinal and let P be any

forcing order such that remains regular in Pgeneric extensions If each

structure has an eeeinsomePextension of the universe then each

structure has a wel lorderedeee in some Pextension of the universe

Prop osition If and each structure has an eeeinsomeconal

itypreserving forcing extension then is weakly Mahlo and is weakly compact

in L

Pro of As is well known if fails to b e weakly Mahlo or fails to b e weakly

compact in L then there is a structure A which has no eee in V We

simply observe that this A continues to work in conalitypreserving forcing

extensions Sp ecically we consider cases

If fails to b e a regular cardinal let cf and set A h g i

where g is a conal map from to and g for Then

A will fail to have an eee in every extension of V

If is regular but not weakly Mahlo let C b e a club which contains

only singular limit ordinals For each cho ose a conal map h

cf Co de these bya mapH by letting H h

when cf C and H otherwise Let A h C H i

Then A will fail to havea wellfounded eee in every extension of V in which

is regular

If is regular but not weakly compact in L then in L there is a

structure A such that A has no eee in any extension of L in which remains

regular

By a similar pro of one may showthatifS is stationary in thenS

is stationary in for some regular This prop ertywas also discussed

in it directly implies the usual weakinaccessible typ e large cardinal

prop erties one gets from a weakly compact cardinal

It is easy to see that every weakly compact cardinal is rveee in every

random real extension so that a rveee cardinal c is equiconsistent with

aweakly compact cardinal In fact if is weakly compact in V and V G

is formed by adding more than random reals or Cohen reals then is

actually veee in V G that is every structure has an eee in some the

universe now V G itself see In particular considering the Cohen

real case we see that the existence of an rveee cardinal cannot haveany

interesting measuretheoretic consequences

AN ABSOLUTENESS RESULT

Corollary ME implies that some c is weakly compact in L

c

Pro of If not then by Prop osition eachuncountable c fails to b e

rveee so by Corollary MA rr holds which is imp ossible

c

So as Carlson notes in by a somewhat dierent argument ME is

c

equiconsistent with the existence of a weakly compact cardinal

An Absoluteness Result

In this section we describ e our absoluteness result for random real mo dels

and use it to prove Theorem As in x we view random real mo dels as

extensions by some measure algebra B The following additional denitions

will b e useful If is a sentence in the forcing language kk denotes its

Bo olean value the maximum condition which forces Each b B has

a countable supp ort suptb which is the minimal subset T suchthat

b is the equivalence class of a cylinder over T If is a B name in the

forcing language supt denotes the union of all suptbsuchthat b is

used hereditarily in the construction of Whenever we name reals or

subsets of wecho ose names with countable supp ort

Theorem Fix such that each uncountable is not rveee

and assume ME Suppose

x Y x Y

B

where is a rst order formula over a structure A Thenitistruein V

that

x Y x Y

Some remarks on syntax The notion of structure is as in x but x and

Y are second order variables here so that equation expresses a sp ecial

prop erty of the cardinal plus whatever function and relation kind of



constants are contained in A Since A is xed here we write instead of

A j If is realvalued measurable then by the metho d of Theorem

holds for all prop erties

n

Of course the theorem holds also for but is trivial in that case by

Sho enelds Theorem In proving the theorem the following denitions will b e useful

AN ABSOLUTENESS RESULT

Denition If Y then Y fx x Y g for

each and Y f x Y g for each x

x

Denition r is the name for the ocial random real That is r

and kr nk ff f ng for n

The notationr is used b oth when we are forcing with some B for

and when we are forcing with a B where is some measure on extending

Leb esgue measure

The following lemma lets us drop a quantier in of Theorem

Lemma Statement of Theorem implies that for some B name

we have supt and Y Y

B B

Pro of Apply the maximal principle to get sucha we can always p ermute

the co ordinates to make supt

In the case that the here happ ens to b e the ocial random realr

Theorem is immediate from the following

Lemma Fix such that each uncountable is not rveee

and assume ME Let be a rst order formula over a structure ALet

Y Assume it is true in V that

x x Y

x

Then for some B name Z Z and

B

r Z

B

extending Leb esgue measure suchthateach Pro of If is any measure on

Y is measurable wemay dene the B name Y fh Y i g

Thus Y and each k Y k Y As a preliminarywe shall

B 

establish

r Y

B 

To conclude this from we need to measure not only the Y but some

sets derived from the formula as well

To simplify the argument we shall measure more sets than necessaryAs

pq usual let L b e the language of A augmented by a constant symbol A

AN ABSOLUTENESS RESULT

for each element of the domain of A For each sentence of L let

A

E fx x Y g Let F b e the collection of all the E where is

x

asentence of L Note that F has size and includes all the Y by using

A

atomic

Toprove from we shall pro duce a measure algebra in whichwe

have k r Y k E for any sentence of L The pro of of k r Y k

A

E will b e by induction on the complexityof but for this pro of to work

the measure algebra will have to preserve certain suprema In particular

each induction step handling an existential quantier gives us a supremum

to preserve For each formula v of L with one free variable v letE

A

S

fE g Note that E E so that a measure algebra

pq vv  pq

satises

E E

vv  pq

W

i it preserves E Apply Theorem to cho ose SP of size such

W

that each measure on which measure FS preserves each E Then

apply ME to x a measure extending Leb esgue measure which measures

FS

Now in the measure algebra B induction on the complexityof do es

indeed showthatk r Y k E for each sentence of L Then in par

A

ticular for the sentence says that E sothat k r Y k

proving

Finallywe conclude by Maharams Theorem Wemay assume that

was chosen so that the measurable sets are generated bythe sets in FS

together with the basic clop en sets in so that the Maharam dimension

of cannot exceed Thus there is an isomorphism i from B onto some

complete subalgebra D of B withiff f ng ff f ng

B  B

for each n and Since i is an isomorphism r Z where

D

Z is the corresp onding D name that is k Dk ik Y k Since is

rst order it is absolute so follows

To nish the pro of of Theorem wemust deal with the p ossibilitythat

the resulting from Lemma is dierent fromr Todothiswe use the

fact that may b e constructed fromr by countable op erations Sp ecically

supp ose x v isan L formula that is mentions the parameter xand

has a free rst order variable v but do es not mention Y Then given the

AN ABSOLUTENESS RESULT

A

structure A and x dene z x by

A

xn A j x n z

Lemma If is a B name supt and then in

V therearea structure A for some countable L and an L formula such

A

that z r

A

Actually need only quantify over so that the map x z xisa

Borel map from to TheL and the A from this lemma have nothing at

all to do with the L and the A from Theorem Nevertheless by merging

the two structures wemay assume that they are the same

Pro of of Theorem Assume and let b e as in Lemma Then

let b e as in Lemma Thus Y r Y where x Y

B

is the formula which asserts z xY Now applying Lemma weget

that condition must b e false for every name Z sothatwe could never

havechosen a Y to satisfy hence

A

Y z xY x

x

which implies

Wenow explain how to derive Theorem from our absoluteness result

We concentrate on the nonpro ductivity of the ccc since the three

parts to Theorem are similar

The following standard construction pro duces a pair of partial orders



whose pro duct is never ccc Start with a graph on vertices

  

Dene the partial orders P and P by setting P fa a is





homogeneous for color g and ordering each P byreverse inclusion Clearly

 

the pro duct P P isnt ccc consider the subset fhfg fgi g



 

Galvin showed that under CH there is a graph such that P and P



are b oth ccc By a dierent argument Roitman showed howtoreado

such a from a random real We use Roitmans construction here

Pro of of Theorem Following x in V injective functions

f for each



For given any x we dene the graph x by

xf gxf for each By forces that each

 r

P is ccc actually just states this for the forcing which adds only

RAMSEY ULTRAFILTERS

the one random realr but then it must b e true also for the B extension

since random real forcing never destroys the cccNow we can co de the f

along with the construction of the x in a suitable structure A and apply

Theorem with Theformula x Y says that either Y is b ounded

 r 

in or Y fails to b e an antichain in one of the P

In Roitman also used similar techniques to get Sspaces and Lspaces

from a single random real and this implies Theorem weomitthe

details which are almost verbatim the same as for Theorem Note

however that to prove Theorem we cannot simply quote the fact that

adding a random real adds an Sspace and an Lspace we use the fact that

the pro of constructs these spaces in a rstorder way from the random real

and some structure in the ground mo del In addition using our absoluteness

result to prove Theorem requires the fact that random real forcing

never destroys Sspaces and Lspaces SimilarlyTodorcevic construction of

an entangled set see page of gives us Theorem Actuallyhe

shows that adding one random real adds an entangled set of size continuum

Thus we get

Theorem Assume ME where and no uncountable is

rveee Then thereisan entangled set of size

Ramsey Ultralters

We b egin by explaining the prop erty of Ramsey ultralters weintend to

refute The following fact is easy to see and was used also in

Lemma Suppose that U is a Ramsey ultralter on

Given a for i thereisanH U such that the sequence

i

ha i H i converges

i

Given a for i with lim a thereisanH U such

i i i

P

that a

i

iH

Of course in once the sum is nite it may b e made arbitrarily small

bycho osing a smaller H

RAMSEY ULTRAFILTERS

Denition A nonprincipal ultralter U on is semiselective i U sat

ises the conclusions and to Lemma

This notion has o ccurred with dierent names in the literature The term

semiselectiveis tak en from where it was p ointed out that under CH

or MA there are semiselective ultralters which are not RamseyWe shall

show that there are no semiselective ultralters if either ME holds or if

the universe was obtained by adding at least random reals to a mo del of



ZF C

We remark that taken separately conclusions and of Lemma

yield twoweaker prop erties of ultralters neither of which can b e refuted in

random real mo dels or from any ME A nonprincipal ultralter satisfying

prop erty is sometimes called rapid There are none of these in the Laver

mo del Miller but when we add random reals any extension of a rapid

ultralter from the ground mo del will still b e a rapid ultralter Satisfying

prop erty is equivalent to b eing a Pp oint and by a result of P E Cohen

there are Pp oints in every random real extension of a mo del of CH

By amalgamating prop erties and we get the following lemma

Lemma Let f beanycontinuous realvalued function on such that

f and f x for x Let U be any nonprincipal ultralter on

Then U is semiselective i for al l thereare H U and

t such that

X

f ji tj

iH

Of course implies lim it since H is innite Now consider

iH

f to b e xed Just in ZF C there is no problem cho osing for each

an innite H satisfying Wenow present a probabilistic argument



wing that the set of all these H can never have the nite intersection sho



prop erty so that there can b e no semiselective ultralters Of course this

argument only works in some mo dels of set theory

The intuition is Cho ose at random and then by some non

random pro cess cho ose H and t so that holds If f is

 

really large then each H must b e so thin that with probabilitywe will



is nite H such that H and H havechosen H

   

 

RAMSEY ULTRAFILTERS

Corollary b elow formalizes this intuition First some notation Let

us use b oth for Leb esgue measure on as well as for the usual pro duct

I

measure on for any set I Let b e a probability measure on a set

X Arandom I sequence indexed by the sample space X is a map

I I

X such that for each Baire set B the set fx B g

x

is measurable and of measure B Note that we are using for x

x

i

Supp ose H X I WeletH fi I x i H g and H fx

x

X x i H g

For the rest of this sectionletf xln x for x

and let f The justication for using this particular f is only that it

makes Theorem true

I

Given any wesay that H I is smal l i for some t

P

f ji tj A smal l process for a random I sequence is a set

iH

H X I such that for each x X H is small and for each i I

x x

i

H is measurable Trivial examples of small pro cesses are or X fig for

any i I

Theorem For any set I ifH is a smal l process for a random I se

P

i 

quence then H

iI

Assuming Theorem for a moment we prove the following two corol

laries the second one immediately implies Theorem

Corollary For any countable set I if H is a smal l process for a random

I sequence then thereare some x y such that jH H j

x y

wehave by Theorem Pro of Computing the exp ectation of jH H j

x y

Z Z Z Z

X

dx dy jH H j dx dy x i y i

x y H H

i

Z Z

X X

i 

y i H x i dx dy

H H

i i

so there must b e some x y such that jH H j

x y

Corollary ME implies that thereare no semiselective ultralters

RAMSEY ULTRAFILTERS

Pro of Let U b e a semiselective ultralter on Let X be with the

usual pro duct measure Let xFor each xcho ose H U suchthat

x x

H is small This denes H X Applying ME let extend the

x x

i

usual pro duct measure on X and measure all the H Then Corollary

yields an immediate contradiction

Later in proving Theorem it will b e imp ortant that Theorem was

stated for X an arbitrary sample space not just Note that Corollary

has a nonvacuous contentjustin ZF C since it is easy to nd Borel

small pro cesses H such that each H is innite

x

WeturnnowtoaproofofTheoremWe rst consider nite pro ducts

n

For each leth b e the largest size of a small set Note that

h is Borel measurable For n let b e the exp ected value of h

n

Z

h d

n

It is easy to see that Also and since

  

every set of size is small Also as n although the growth

n

rate is fairly small as we shall show in Lemma First we use the size of

the to place a crude upp er b ound to the sum in Theorem

n

Lemma Let I be any set and let H be a smal l process for a random I

P P

i  

sequence Then H m

m

iI

m

Pro of It suces to prove this for I nite and it is trivial if I is emptyso

say jI j n Then wemayaswell assume that I is the ordinal n

 n

arranged so that H H H For each m n H m

x

is m small so jH mj h m Then

x x x

Z Z

X

i

H jH mj dx h m dx

x x

X X

im

Z

h d

m

m



Setting m j

X

i j

H m H

m

m

im

i

since H as i and the result nowfollows by summing over j

RAMSEY ULTRAFILTERS

 r

Lemma Whenever r n r n e e

n

Pro of First note that f is p ositiveand f is negativeon which

implies that f is increasing on and f a b f af b whenever

a b a b Also the inverse of f is a smallexp onen tial If

 y

y ln x then x e e

n

Now x n r and let S f h r g Then

n

r nS To estimate S x S and then x H n suchthat

P

jH j r and H is small Then x t suchthat f ji

iH

j j tj List H as fi j i j gFor r f ji

r r

f ji tj f jj tj Since the sum is there is an suchthat

f ji j j r So wehave shown that for each S there are

distinct i j such that f ji j j r

n

Now for each particular i j f f ji j j r g

n

 r 

e e Since there are n p ossibilities for i j wehave S



  r

n e e proving the lemma

Of course for some rn the estimate in Lemma is worse than the

obvious n but bycho osing an appropriate r for each nwemay obtain

n

an upp er b ound which is sucient to prove the desired result

Pro of of Theorem By Lemma it is sucienttoprovethat

P



n Todothiswe break the sum into blo cks where blo ck

n

n

r

sums from n to and blo ck r forr sums for n to

r

On eachblock we apply Lemma

 

On blo ckFor n wehave e e

n

using r which implies that





X

 

n

n

n

On blo ck r for r r so

n

r





X X

r

 

n r

n

 r

n

r r

n   n  

and it is easy to see that



X

r

r

r 

RAMSEY ULTRAFILTERS

Wenow add forcing to this pro of and show that there are no semiselective

ultralters in any mo del obtained by adding or more random reals We



continue the notation of xx forcing with B We continue to use to

denote Leb esgue measure on or anypower thereof and to denote the

usual measure on anypower of f g

Denition For any set I arandom I sequencename is a name in

the forcing language such that k I k andforeach Baire

I

B k B k B

Note that in the formalism of forcing inside the kk I should really

be I whereas B should really b e a Borel co de see not B

Lemma Suppose that and arerandom I sequencenames and sup



pose that and names in the forcing language such that each is forced



by to be a subset of I and to be smal l Assume that supt

supt is disjoint from supt supt Then some forcing condition p

 

forces j j



Pro of This lemmainvolves forcing so it ma y b e p erceived to take place in

some ground mo del V The pro of is more transparentifweviewV as b eing

countable from the outside so wemay refer explicitly to generic ob jects Let

Z In the case of random real forcing it is more convenient to think

of the generic ob ject as an ob ject z Z whichisrandom over V that is

not in any V co ded Baire nullset rather than a generic lter see To

prove the lemma it is sucient to nd some z random over V suchthat

jval z val zj where val refers to the value of a name in the



generic extension

T nT

Let T supt supt let Y and let X Then we



mayidentify Z with X Y and think of the extension V z V x y

as the iterated extension V y x Fix a y Y which is random over V

Then for all x X x is random over V y ix y is random over V

Furthermore x is random over V y for almost every x X For let

val x y Since supt is disjoint from T is a random I

x

sequence Dene H X I so that H val x y when x is random

x l

over V y and H otherwise Then H is a small pro cess for For

x

RANDOM GRAPHS

i

each i I suptki k suptki k T which implies that H and



i

H are sto chastically indep endent Now applying Theorem and the



CauchySchwarz inequality

Z

X X

i i i i

jH H jdx H H H H

x x

 

iI iI

X X

 

i  i 

H H



iI iI

Hence fx X jH H jg so wemaycho ose an x random

x x

over V y suchthat val x y val x y H H has size or

 x x

as required

Now Theorem follows immediately from

Theorem Suppose that Then it is forcedby that thereareno



semiselective ultralters in the generic extension by B

Pro of If the theorem fails then by homogeneityof B there would b e a

name U such that forces U to b e a Ramsey ultralter In V x disjointly

supp orted random sequencenames for Thenxnames such



that forces that U and that is small By the usual pressingdown

argument wemay nd distinct suchthat supt supt is disjoint

from supt supt Then contradict Lemma

Random Graphs

In this section we showhowtoderive Theorems and directly from a

result in random graph theory ab out partitions on nite sets For background

in this sub ject see



The intuition is for a xed set I cho ose a partition I at

random and then by some nonrandom pro cess construct a homogeneous

set H for There are many results in the literature going backtoa

pap er of Erdos to the eect that with high probability H must b e fairly

thin Erdos used this to establish an exp onential lower b ound for the

Ramsey numb ers

To formalize this intuition we use the following general framework Let

X b e a probability space A random partition of a set I indexed by

RANDOM GRAPHS

X is a map such that for each x X ie x is a partition

x x

 

I and such that the sets E fx i j gfor fi j gI

ij x

are measurable and are indep endentevents of probability ie measure



A homogeneous process for is a set H X I such that for each

i

x X H is homogeneous for and for each i I H is measurable

x x

Trivial examples of homogeneous pro cesses are orX fi j g for any i j I

The sp ecic theorem we need is

Theorem If H is a homogeneous process for a random partition of a

P

i 

H set I then

iI

Note that Theorem do es not assume I to b e nite although once we

prove it for nite I it follows immediately for all I From Theorem we

may derive Theorems and exactly as in x we omit the details of this

Before proving Theorem wemention the following corollary in nite

combinatorics Roughlyifwecho ose homogeneous sets for partitions using

any nonrandom pro cess then there is a non probability that twoofthe

homogeneous sets will have small intersection Formally

Corollary Assume that I is countable or nite isarandom par

tition of I indexedbyX and H is a homogeneous process for Then

fx y jH H jg

x y

R R

Pro of Let E dx dy jH H j that is the exp ectation of jH H j

x y x y

Let w fx y jH H jg Then E w But also

x y

exactly as in the pro of of Corollary wehave E Combining these

wegetw

We remark that the corollary is probably interesting primarily for the



I

sp ecial case of the naturalrandom partition where X is with the

usual pro duct measure and is just xbutwe seem to need the more

x

general statement of Theorem to derive Theorem and the pro of of

the general statement is no harder than the pro of of the sp ecial case

We pro ceed nowtoprove Theorem Actuallywe susp ect that our

result is not b est p ossible in that p ossibly the in Theorem could

b e replace by whichwould mean that the in Corollary could

RANDOM GRAPHS

b e replaced by However our pro of involves a sequence of estimates

eachoneintro ducing a bit of slop in the nal result

First the following lemma can b e used to b ound a sum of squares

Lemma Suppose a a bb arereal numbers such that

 n  n

a a b b for j n and a a a

 j  j  n

   

Then a a b b

 n  n

Pro of Wemay assume that each b otherwise replace b by jb j Since

j j j

the lemma is trivial for n we pro ceed by induction So x n and x

nonnegative b and assume that the lemma holds for all smaller values of n

j

Now by compactness x numbers a a satisfying which maximize

 n

 

a a

n 

If for some jnwehave a a b b thenwehave

 j  j

a a b b for k j n so applying the induction

j  k j  k

   

hyp othesis a Since the induction hyp othesis b b a

j  n j  n

       

also implies that a a b b wehave a a b b

 j  j  n  n

So assume that a a b b for each jnWemust

 j  j

otherwise we could replace each a then have a a b b

n i  n 

by some a and contradict maximality But then a so for a small

i n

enough the sequence a a a a satises andcontradicts

  n n

   

maximality since a a implies that a a a a

 n  n

 n

We shall prove Theorem by using this lemma plus a crude upp er



 n 

b ound Lemma on the partial sums of the H Let G and

i n

let denote the usual counting probability measure on G Wemay think

n

of elements of G as random graphs or partitions on n no des since each

n

G is a partition of the pairs from the nelement set n fn g

n

into pieces For eachG leth b e the largest size of a homogeneous

n

set for For n let b e the exp ected value of h

n

Z

h d

n

It is easy to see that Also and

   

since every set of size or less is homogeneous It is not hard to see by direct

computation Lemma that and It is wellknown



RANDOM GRAPHS

that for large n is approximately lg n where lg xis log x

n 

A suitable upp er b ound on the can b e used to prove Theorem by

n

applying the following lemma

Lemma Suppose that for each m s with s As in Theorem

m m 

let H be a homogeneous process for a random partition of a set I Then

P P

i  

H s s

n n

iI

n 

Pro of As remarked ab ove it is sucienttoprove this when I is nite and it

is trivial if I is emptysosay jI j n Then wemayaswell assume that I

 n

is the ordinal n arranged so that H H H For each

 

m n H m is homogeneous for m sojH mj h m

x x x x

Then

Z Z

X

i 

H jH mj dx h m dx

x x

X X

im

Z

h d s

m m

G

m

i

Let a H and b s s Setting m j for j nwe

i i i i

have a a s b b since s so the result follows

 j j   j 

by Lemma

Using the approximation lg n together with the fact that

n

P

i 

H wecannowbound log n log n log

iI

n n

P





by something like However to achieve the b ound



n

log  n

of wemust work a little harder For small nwe use the exact value

n

rather than the asymptotic approximation

Lemma

  

Pro of For G has size By insp ection of the G have

h and of the G have h

For G has size By insp ection of the G have h

  

of the G have h and of the G have h



For larger nwe compute an upp er b ound on as follows For r n

n

n

we dene f G h r g

n r

RANDOM GRAPHS

Lemma a For rn

n n

r n r

n

r r 

b For r n

n



rr  n

r

r

Pro of For a we partition G into three pieces where h r h r

n

n n n n

and h r This yields r r n

n

r r r  r 

For b note that h r i there is some A n of size r which

is homogeneous for Assume r since otherwise b is trivial since

n

n

Then there are exactly p ossibilities for A and each A can b e

r

r

homogeneous byhaving all its pairs colored either or so the probability

r



that A is homogeneous is exactly

These estimates on are not quite as crude as they may seem at rst

n

sight since it is known that for large n there is some r lg n

such that mostof the G have h equal to r or r so that r

n n

n

For this r the estimates in Lemma will compute n to b e negligible

r 

b ounding by some value near r Itwas our use of at all in Lemma

n n

that was real ly crude

Lemma

  

   



Pro of We just cho ose the value the r which yields the b est estimate on

n

by Lemma For n weuse r for n we use r

for n weuser For n we use Lemma instead

Calling the estimate in Lemma s as in Lemma for n

n

P



and setting s weget s s Note that

 n n

n 

replacing the exact b e the larger yielded a smaller by value

of the sum Let lg s so s lg To estimate

 

the tail of the series it will b e sucienttoprovethat lg n for

n

all n To do that we use the following estimate

RANDOM GRAPHS

Lemma Suppose that r is an integer with rnand suppose

r lg n a where a Then

a r

r

n

r

p

r r ra

r r e Since n and Pro of By Stirlings Formula r

n

r

n r Lemma implies

r

r

e

rar  n

p

r

r

r

Of course the same estimate holds if we replace r by r and a by a

r 

Wemay simplify this by observing that r e for all r take

r  r r 

the log of b oth sides It follows that er e r so

r

e

n ar 

p

r 

r r

r

ra

Since n

r

e

n ar ra

p

n

r 

r r

r

n n

Since by Lemma r n

n

r r 

r

e

a rar 

p

r

n

r r

r

Now r implies that r e and r so

a rar 

r

n

r

Finally using a yields the desired result

Lemma If n then lg n

n

Pro of We apply Lemma Cho ose a suchthat a and suchthat

r lg n a is an integer Then r lg n and r

n

 

ADDITIONAL REMARKS

Pro of of Theorem First note that for all n lg n

n

For n this follows from Lemma For smaller nwe just compute

it using Lemma using r for n r for n

r for n r for n r for n and

r for n

Thus we can in fact set s lg n for all n as indicated

n

ab ove whence applying Lemma

X

i 

H

iI

X X

 

s s s s

n n n n

n  n 

X

 

log n log n l og

n 

X



l og



n

n 



l og

Additional Remarks

Wepoint out here that at least for small our results ab out ME are b est

p ossible

First we note that in the axiom ME one cannot replace byan

arbitrary measure space For example Theorem of Rao consider the

measure space where countable sets have measure cocountable sets

have measure and other sets are not measurable Then no extension of

can measure the countable collection of sides of rectangles whose generated

algebra contains the wellorder on since such a measure would contradict

Fubinis Theorem By a related use of Fubinis Theorem Grzegorek

shows that ME X must fail some atomless measures Sp ecicallyif

is the least size of a non Leb esgue measurable subset of the real line and

X isany atomless probability space with jX j thenME X is

false The following lemma and corollary also use this metho d of pro of

ADDITIONAL REMARKS

Lemma Suppose Y and suppose Y Y as where

is some limit ordinal and each Y is a Lebesgue nul lset Then thereisa

countable family E of subsets of such that every measure extending

Lebesgue measure which measures al l the sets in E makes Y a nul lset

Pro of First x Let b e Leb esgue measure on For each

y

cover Y by an op en set U withU If y Y let V

y

be U where is least suchthat y Y Ify Y letV Now we

y

have constructed a V with every horizontal slice V an op en

set of measure Let fB n g b e an op en base for and let

n

y

A A fy B V gLet be any extension of such that each

n n n

A is measurable Then V is in the algebra generated by measurable

n

R

rectangles and hence is measurable so V d x byFubinis

x



Theorem Now for x Y the vertical slice V contains all of Y except for

x

R



a Leb esgue nullset so V d x Y where is outer measure

x





Hence Y

i

Now was arbitrarysowemayletE fA n i g

n

Of course Lemma is trivial unless ME is true

Corollary ME implies that no Lebesgue measurable set of positive

measureisanincreasing union of Lebesgue nul lsets

Next wemake some remarks on the additivities of our measures



Theorem For any cardinal there is a family E of subsets of



with the fol lowing property Whenever is a probability measureon such



that each set in E is measurable and each singleton is a nul lset then is

a union of nul lsets



Pro of Note that if we allowed E to havesize there would b e an obvious

Ulam matrix argument here but to get E of size we need a bit more

care



For each letR wellorder in typ e at least Let E be the



family of all sets of the form f R g where Fix a

measure as ab ove By the standard exhaustion argument it is sucient

to nd a union of nullsets whichcovers some measurable set of p ositive

REFERENCES

measure so we assume that this never happ ens and derive a contradiction

This argumentmaybeviewed either as an attempt to pro duce the Ulam

matrix by just measuring sets or as an attempt to apply Solovays

Bo olean ultrap ower technique to the R to pro duce in some random real



extension a wellorder of of typ e

S



Let I b e the family of all X suchthat X N for some

sequence N of nullsets By our assumption on nomeasurable set

of p ositive measure is in I Clearly I is an ideal and every union of

elements of I is in I



Let b e the family of all X suchthat X B I for some



measurable B Observe that is a subalgebra of P and is closed under

unions and intersections Dene a measure on so that X B

for some measurable B with X B I note that this denition of X



is indep endent of the B chosen Also note that is additive in the sense

S P

that X X whenever the X are disjointsetsin In



particular every prop er initial segmentof has measure

 

For and let E f rank R g



measurable Now the E By induction on prove that each E is

form a measurable Ulam matrix which yields an immediate contradiction

In Carlson proves this for the case cFor this case his argument

is much simpler than ours He notes that the rows of the Ulam matrix are

disjoint so eachrow can b e countably generated bycountably many sets so



the entire matrix can b e countably generated by sets

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