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Set Theoretic Properties of Loeb Measure 2

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1 Arnold W Miller University of Wisconsin Madison WI millermathwiscedu 2 Set theoretic prop erties of Lo eb measure Abstract In this pap er we ask the question to what extent do basic set theoretic prop erties of Lo eb measure dep end on the nonstandard universe and on prop erties of the mo del of set theory in which it lies We show that assuming Martins axiom and saturation the smallest cover by Lo eb measure zero sets must have cardinal ity less than In contrast to this we show that the additivity of Lo eb measure cannot b e greater than Dene cof H as 1 the smallest cardinality of a family of Lo eb measure zero sets which cover every other Lo eb measure zero set We show that H car dblog H c cof H car d where car d is the external 2 cardinality We answer a question of Paris and Mills concerning cuts in nonstandard mo dels of number theory We also present a pair of nonstandard universes M and N and hypernite integer H H M such that H is not enlarged by N contains new el ements but every new subset of H has Lo eb measure zero We show that it is consistent that there exists a Sierpi nskiset in the reals but no Lo ebSierpi nskiset in any nonstandard universe We also show that is consistent with the failure of the continuum hy p othesis that Lo ebSierpi nskisets can exist in some nonstandard universes and even in an ultrap ower of a standard universe Let H b e a hypernite set in an saturated nonstandard universe Let 1 b e the counting measure on H ie for any internal subset A of H let A jAj b e the nonstandard rational where jAj is the internal cardinality of A a jH j hyperinteger Lo eb showed that the standard part of has a nat ural extension to a countably additive measure on the algebra generated 1 Partially supp orted by NSF grant AMS classication H A A 2 in Journal of Symbolic Logic by the internal subsets of H For more background on nonstandard measure theory and its applications see the survey article Cutland Refer ences on general prop erties of Lo eb measure and the algebra generated by the internal sets include Henson and Keisler et al Some prop erties of Lo eb measure are the following For every Lo eb mea surable set X there exists an internal set A H such that X A A X has Lo eb measure zero Also every Lo eb measure zero set can b e covered by one of the form A where each A is an internal set of measure less than n n n 1 It is related to Leb esgue measure on the unit interval Identify H n+1 1 Let st T with the time line T fnt n jH jg where t jH j b e the standard part map Then for any Leb esgue measurable Z the 1 set st Z is Lo eb measurable with the same measure as Z Theorem KeislerLeth If F is a family of internal Loeb mea sure zero subsets of an innite hypernite set H the nonstandard universe is saturated and F has external cardinality less than then there exists an internal set A H of Loeb measure zero which covers every element of F pro ofConsider the sentences A jK j fB A B F g f n g jH j n where A is a variable Clearly this set of sentences is nitely satisable and has cardinality the same as F It follows by saturation that some internal A satises them all simultaneously 2 Note that this implies that in a saturated universe any external X H of cardinality less that has Lo eb measure zero Theorem Suppose the universe is saturated and every set of reals of 1 cardinality has Lebesgue measure zero then for any innite hypernite set H and X H of an external cardinality X has Loeb measure zero pro ofThe standard part map shows this 2 In this case it may b e imp ossible to cover X with an internal set of Lo eb measure zero Theorem For any M an saturated universe and H an innite hy 1 pernite set in M there exists any elementary saturated extension N of 1 M which has the property that there exists a family fA H g of 1 internal Loeb measure zero subsets of N such that for every internal Loeb measure zero B H in N there exists with B A 1 pro ofBuild an elementary chain of saturated universes M for 1 1 Use the pro of of Theorem to get A M such that every Lo eb measure +1 zero set B M is covered by A 2 By using a simple diagonal argument in a universe given by the last theorem there will b e an X H of cardinality which is not covered by 1 any internal Lo eb measure zero set In fact the set X will have the prop erty that for every internal Lo eb measure zero A H X A is countable We say that X H is a Lo ebSierpi nski set i it is uncountable and meets every Lo eb measure zero set in a countable set Thus what we have here is a weak kind of Lo ebSierpi nski set If MACH is true then every set of reals of cardinality has measure zero So by Theorem this weak 1 Lo ebSierpi nski set would not b e a Lo ebSierpi nski set Now we ask for the smallest cardinality of a cover of H by Lo eb measure 1 zero sets Note that since monads st fpg for some p have measure zero H can always b e covered by continuum many Lo eb measure zero sets MA stands for the version of Martins Axiom which says that for any par tially ordered set P which has the countable chain condition and any family D of dense subsets of P of cardinality less than there exists a Plter G which meets all the dense sets in D Theorem Suppose MA and the nonstandard universe is saturated then H cannot be covered by fewer than sets of Loeb measure zero pro ofLet P b e the partial order of all internal subsets of H of p ositive Lo eb measure ordered by inclusion where stronger conditions are smaller Forc ing with P is the same as using the measure algebra formed by taking the algebra generated by the internal subsets of H and dividing out by the Lo eb measure zero sets It has the countable chain condition b ecause it is 1 imp ossible to nd n sets of measure greater than which have pairwise n intersections of measure zero T For let f K g b e a family of many Lo eb measure n n zero sets where each K is an internal subset of H of Lo eb measure less than n 1 For each dene a dense D P by n+1 D fb P n b K g n Using MA let G b e a Plter meeting every D for For each let n G Consider the family of sentences x n b e such that H K n g fx H K Since G is a Plter this family of sentences is nitely satisable Hence by saturation some internal x H satises them all But this shows that the family of Lo eb measure zero sets did not cover H 2 This partial order was also used in Kaufmann and Schmerl Next we consider the additivity of Lo eb measure Here we show that it is always as small as p ossible Theorem For any innite hypernite H in an saturated universe 1 there exists a family of Loeb measure zero sets whose union does not have 1 measure zero K pro ofWithout loss of generality we may assume H for some internal hyperinteger K since H may b e replaced by its internal cardinality and for K K +1 K some hyperinteger K we have H and has Lo eb measure at least in H Supp ose K where K is the innite countable subsets of K and let f n g For each n dene n K H fh h h h g n 0 1 n1 1 Then each H is an internal set of measure n n 2 T Lemma If A is an internal set and H A then there exists n n n such that H A n This follows from the fact that the H are a descending sequence and the n universe is saturated 1 2 Let K for b e a family of disjoint countable subsets of K 1 Let H b e dened as H was but using instead of Dene A H n n n n So each A has Lo eb measure zero We show that the union A cannot 1 have Lo eb measure zero Supp ose A is an internal set and A A 1 By the Lemma there exists n and f m g such that m 1 m m for every m H A Let K H Note that each K for m m n n 1 m has Lo eb measure and they are indep endent ie for any n 2 m m m the Lo eb measure of 1 2 k K K K m m m 1 2 k k is Consequently K has Lo eb measure one since the measure of m m N N H K is and as N Consequently the mN m Lo eb measure of A is one and the theorem is proved 2 The last cardinal asso ciated with Lo eb measure we will consider is the conality cof H This is the smallest cardinality of a family F of Lo eb measure zero sets such every Lo eb measure zero set is covered by a member of the family F Theorem Let car d be the external cardinality function and H an innite hyperinteger in some saturated universe then 1 H car dblog H c cof H car d 2 pro ofThe rst inequality is proved by a similar argument to the pro of of the K K +1 last theorem Let the hypernite integer K blog H c so H 2 Let f car dK g b e disjoint countable subsets of K and dene A H as ab ove The argument ab ove shows that no Lo eb measure zero set covers uncountably many of the A hence car dK cof H The second inequality follows from the fact that every Lo eb measure zero set is covered by a Lo eb measure zero set in the algebra generated by the internal subsets of H and the following result of Shelah if L is an innite hypernite set in an saturated universe and car dL then 1 H take L the number of internal subsets of H 2 In Keisler it is shown under GCH that for any set of innite successor cardinals C there exists a nonstandard universe M in which C fcar dH H is a hypernite set in M g For nite C the nonstandard universe M can b e an ultrap ower of a standard universe In Shelah it is shown that for any countable theory T with distinguished unary predicates Q and P if for every n T has a mo del M n M n M M such that n jQ j jP j then T
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