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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