2 1. ELEMENTAR Y EMBEDDING S 1.1.1 Example . Le t N b e th e natura l number s an d an d le t U b e a n ul - trafilter o n N suc h tha t fo r eac h n G N, {n} 0 U. Le t j : V - V N /U b e the induce d embedding . Th e ultrapowe r V N /U i s a typica l exampl e o f a nonstandard model fo r example , i f / : N N i s th e identit y functio n the n [f]u £ J W i s a nonstandar d integer . An ultrafilte r U i s principal i f it ha s a membe r wit h exactl y on e element , otherwise i t i s calle d nonprincipal. If U i s principal , the n th e transitiv e collapse o f th e ultrapowe r o f an y structur e b y U i s identica l t o th e origina l structure, an d th e compositio n o f th e transitiv e collaps e wit h correspondin g mapping i s th e identit y mapping . If n i s a cardina l an d U i s a n ultrafilter , the n U i s K-complete i f for ever y 7 K an d ever y se t {A a : a 7 } C U, th e intersectio n njAo , : a 7 } is i n U. Th e completeness o f a n ultrafilte r U i s th e leas t cardina l K, such that U i s no t K + -complete. If U i s principal , the n ther e i s n o suc h K,, an d so b y convention , i f w e specif y th e completenes s o f a n ultrafilter , the n th e ultrafilter i s presume d t o b e nonprincipal . O n th e othe r hand , th e clas s o f ^-complete ultrafilters , fo r an y cardina l « , include s th e principa l ultrafilters . Ultrafilters whic h ar e cji-complet e ar e calle d countably complete. 1.1.2 Exercise . Sho w tha t i f U i s a countabl y complet e ultrafilte r o n a set X , an d th e Axio m o f Choic e holds , the n V x /U i s wellfounded . (Hint : Suppose tha t th e function s fi: X -* Ord, (i uo) represent th e member s of a descendin g sequenc e i n V x /U', an d fo r eac h pai r o f integer s i j , let Aitj = {xeX I fi(x) fj(x)}.) If X i s a se t an d \i\ V(X) R i s a function , w e sa y tha t / i i s uniform if /i({x} ) = 0 fo r al l x G X , finitely additive i f fo r an y finit e sequenc e A i , . . . , A n o f pairwis e disjoin t subset s o f X , M U ^ ) = E M^) , lz n lz n and countably complete i f fo r an y se t { ^ : i a } C V(X), i f /i(-Az ) = 0 for eac h i CJ , then /i(Uio ^ ) ~ 0 . ^ o r o u r purposes , a measure o n a n infinite se t X i s a finitely additiv e functio n /i : V(X) » [0,1] wit h /z(X ) = 1 . In 1930 , Banac h [2 ] aske d whethe r ther e coul d exis t a unifor m countabl y complete measure . Thi s questio n le d Ula m [49 ] t o isolat e th e notio n o f measurable cardinal . W e refe r th e reade r t o Chapte r 1 o f [21 ] fo r mor e o n the discover y o f measurabl e cardinals . 1.1.3 Definition . A cardina l K is measurable i f K is uncountabl e an d ther e exists a ^-complet e nonprincipa l ultrafilte r o n n. We wil l conside r onl y measure s wit h rang e {0,1} . Fo r suc h measures , the preimag e o f {1 } is an ultrafilter , an d w e will use th e term s completeness,
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