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Weak Measure Extension Axioms * 1 Introduction

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Weak Measure Extension Axioms * 1 Introduction 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
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