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Arnold W Miller
University of Wisconsin
Madison WI
miller math wisc edu
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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 Martin s 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 De ne cof H as
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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 d blog H c cof H car d where car d is the external
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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 hyper nite 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 eb Sierpi nskiset in any nonstandard universe We
also show that is consistent with the failure of the continuum hy
p othesis that Lo eb Sierpi nskisets can exist in some nonstandard
universes and even in an ultrap ower of a standard universe
Let H b e a hyper nite set in an saturated nonstandard universe Let
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