THE JOURNAL OF SYMBOLIC LOGIC
Volume 63, Number 2, June 1998
ULTRAFILTERS WHICH EXTEND MEASURES
MICHAEL BENEDIKT
Abstract. We study classes of ultrafilters on m defined by a natural property of the Loeb measure in the
Nonstandard Universe corresponding to the ultrafilter. This class, the Property M ultrafilters, is shown
to contain all ultrafilters built up by taking iterated products over collections of pairwise nonisomorphic
selective ultrafilters. Results on Property M ultrafilters are applied to the construction of extensions of
probability measures, and to the study of measurable reductions between ultrafilters.
§1. Introduction. Many classes of ultrafilters have been studied that can be characterized by the properties of the ultrapowers of arithmetic which can be formed
using them (see, for example, [3] [16]). This paper studies a class of ultrafilters on co
that is nicely characterized in terms of nonstandard analysis. In §3 we define Property M ultrafilters, and show how they can be constructed. Results of Henson and
Wattenberg [11] show that this class contains the selective ultrafilters. The principal
results of §3 are that products of nonisomorphic selective ultranlters have Property
M, and that taking iterated products over nonisomorphic selective ultranlters gives
a large class of Property M ultranlters . These constructions are applications of
the techniques in Blass [5]. The independence of Property M ultranlters has been
recently proved by Shelah [19]: this and related results are discussed at the end of
§3. In §4 we give some applications of these results. First, we introduce a way of
'pushing down' the Loeb Measure construction of nonstandard analysis down to a
standard measure on the original sample space. This extension process was used by
the author to study the properties of Loeb measure in nonstandard universes formed
via selective ultrafilters; a similar one was independently studied by Fremlin [9] to
prove a result in the theory of random graphs. We use the results of §3 to answer a
number of questions about these extension measures. We show that the extensions
given by different selective ultrafilters must be compatible, and that, under MA,
there is a measure containing all extensions via selective ultrafilters. We also show
how the results of §3 can be used to answer questions on measurable reducibility of
ultrafilters.
§2. Preliminaries. All ultrafilters considered here will be over a countable set.
Whenever the structure of the index set isn't crucial, we will take the index set to be
co. We assume familiarity with the basic properties of the selective, semi-selective,
F'-point, Q-point, and rapid ultrafilters [6]. [6] also provides basic facts about the
Received April 18, 1995; revised February 25, 1997.
© 1998, Association for Symbolic Logic
0022-4812/98/6302-0022/S3.50
638
ULTRAFILTERS WHICH EXTEND MEASURES
639
Rudin-Kiesler order, denoted here by <RK- For ultrafilters U on X and a sequence
( W,• : i £ co) of ultrafilters on Y, we let J2u ^ denote the sum ultrafilter whose
elements are
xY:{n:Sn£Wn}£U},
{S cX
where Sn = {m : (n,m) £ S}.
For ultrafilters U on X and a f f o n K the product U x W is ^ ^ W,, where each
Wj = W. We also refer to [6] for facts about products of ultrafilters.
We also assume familiarity with the construction of nonstandard universes from
ultrafilters, and with the Loeb measure construction [1]. The following paragraph
reviews parts of the construction necessary to establish the notation used in this
paper.
Let S be any set of urelements, and let V^iS) be the coth level of the iterated
heirarchy based on S, i.e.,
V0(S) = S,
V„+i(S) = Vn(S)UP(Vn(S)),
Va,(S) = ( J Vn(S).
Any model of the form (VW(S),£) is called a superstructure. Given an ultrafilter
U, we can form the following triple:
(v0J(s),e) -^ (J]
VW(S),E\
M
(*V,G)
where j is the Mostowski collapse map with domain the well-founded part of
(Ylu Vm{S),E), du(x) = (x,x,...) c/ ,and*Fisthesuperstructure(F (U (;(J c/ (5*))).
The object *V, or *VV when we want to emphasize which ultrafilter was used,
is called the nonstandard universe (or NSU) formed via U over the superstructure
Vm(S). The map j o d is usually denoted by "*". In sections where there is only
one nonstandard universe mentioned, we will use *V to denote the NSU, and the
usual "*" to denote the map j o d. In sections where we need a notation for the
nonstandard objects that tells something about how they were constructed from the
ultrapower, we will refer to the NSU as FJt/ Vm{S), and use dv(x) to mean j o dv.
More generally, we will use the following convention:
Let our ultrafilter U be over index set A. For any model of the form (S, e), we
will use rii/(^' e ) t 0 denote the Mostowski collapse of the well-founded part of the
ultrapower, and we will use dv(x) to denote the object j o dv{x) in the collapse.
For (A()j€oj a sequence of sets in V, (Aj : i £ A)u will mean the image under
the Mostowski collapse map j of the equivalence class of (Ai}ieA in the ultrapower.
Thus a statement like:
du(B)n(Ai:i£A)u^0
makes sense as a statement about the collapse.
Sets of the form (Aj : i £ A)v are called internal sets while all other sets in *V
are called external. Sets in the image of the *-map are the standard sets.
We will use similar notation for ultraproducts: \[u(Si, e) will denote the Mostowski collapse of the well-founded part of the ultraproduct. For x such that
{i : x £ Sj) £ U,
640
MICHAEL BENEDIKT
du(x) will denote the image under the Mostowski collapse map of the equivalence
class of the function whose value is x a.s. U.
For elements of the base set S of the superstructure Vm(S) the *-map is the
identity, so for b e 51 we can write simply b rather than dv (b) or *b: we will usually
assume that N and 5ft are subsets of the base set, and thus we can identify r e3J
with *r. We will do the same for some frequently used operations on 5ft and on TV,
such as < and +.
For a nonstandard real number we will use °r to denote the standard part of R.
For a *-fmitely additive measure fi, Lfi denotes the Loeb measure formed over /u
(see [14]).
We will often deal with the Loeb extension of a standard finitely-additive measure
*ju. In this case we will always write Lju, rather than L*ju, for the extension. When
it is necessary to indicate which ultrafilter U is used in forming the Loeb measure,
we will write Lufi.
A selective universe will mean an NSU formed using a selective ultrafilter. Similar
notation will be used for universes formed via other classes of ultrafilters.
Finally, the following notation is used throughout this paper:
For a set 5" in *V we denote by a(S) the set { x e V : *x e S }. If *V is formed
via the ultrafilter U with index set A, and S = (S,: : i e A ) v is an internal set in
*V, theno-(S') is just
U-limS, = {x:{i
: x e S , } e £/}.
Let {X, £, fi) be a probability space. A family of sets F is said to be a compact
class if every subfamily of F having the finite intersection property has nonempty
intersection. A a-algebra T c S is said to be compact if there is a compact class
F such that every set in T can be approximated from within (in inner measure) by
sets in F. (X,Y,,fi) is perfect if for every countably-generated sub-er-algebra T of I,
T is compact. Perfect measures are discussed in [18]. The class of perfect measures
includes all Radon measures, and hence all Borel measures on a compact metric
space.
§3. Property M. For this paper our ultrafilters will always be over a countable
set, which will be taken to be co whenever the structure of the set isn't important (as
in the definitions below).
DEFINITION. An ultrafilter has Property M if: whenever (X,I.,fi) is a perfect
probability space, and A is a set in the NSU formed via U such that Lfi{A) > 0,
then a U ) ^ 0 .
That is, U has Property M if, in *VUt sets of nonnegligible measure have standard
elements.
Since the definition of Property M seems a bit cumbersome, we proceed to give
some equivalent formulations of Property M that are easier to check. To verify
Property M, it clearly suffices to consider internal *-measurable sets in *VV. Since
the internals all have representations within the ultrapower, Property M reduces to:
Whenever (X,T.,ju) is perfect, and (^4,),ea) is a sequence of/i-measurable sets
with measure bounded away from 0, then 3S e U such that f]ieS At ^ 0.
641
ULTRAFILTERS WHICH EXTEND MEASURES
Let A be Lebesgue measure on [0,1], with c-algebra Y,\. Let Z™+ be the set of
sequences (Aj)i&m such that Be, V7, X(At) > e > 0. As a further reduction we have:
LEMMA 1. Suppose that U has the property that:
for every sequence {Aj)i€aj
G £™+ BX G U
such that
j | A/ ^ 0.
iex
Then U has Property M.
PROOF.
(f)
We first show that the hypothesis implies that:
\/{Ai)iew
G rf + BSeU
such that
If this were not the case for some (Ai)iew,
Vs e [S]<w X \C\ A \ > 0.
let
T=Le[co]<OJ:x(f]Al)
=o|,
and let B — \JseT f]ies At. Then if A is the set (At : / G co)u, we have LX{A) > 0,
a {A) C B, and X{B) = 0 . We have LAU ~ *B) > O.butthereisnoxincrU ~ *B).
Take an internal set A' of positive Loeb measure inside A ~ *B. Then ^4' = (A'^u
with (y4j) G 2™+, and a {A') = 0, and this contradicts the hypothesis.
Now let (Bj)iew be a sequence of JU-measurable sets for some perfect measure
(X, S, fi) such that Be Vi //(5,-) > e > 0. Without loss of generality we can assume
that the fi, s are members of some compact class. Let T be the <r-algebra generated by
the BjS, let (M, ju) be the measure algebra associated to the measure space (X, T, //),
and let n: T ==> M be the mapping that sends a measurable set to its equivalence
class in the measure algebra. We apply the following basic fact from measure theory,
a simple consequence of Maharam's Theorem (see, for example, Corollary 3.12 of
[10]):
PROPOSITION 1. For every countable generated probability measure algebra (M, p)
there is a measure-preserving homomorphism f from M into the measure-algebra ofX.
Let / be such a measure-preserving homomorphism, and let C, be a A-measurable
set whose equivalence class is f{n{Bi)).
Since / is measure-preserving, we have
X(Cj) > e, and our hypotheses then imply, via (f), that
BS eU
such that
Vs G [S]<0} p j C, ^ 0.
But this implies that we have
S eU
such that
Vs G [S]<0J M (pj B-\ > 0,
and since the B, are in a compact family, we have f]i€S B, ^ 0.
H
Henson and Wattenberg showed that
THEOREM 2 ([11]). If U is selective, (X,rr,fi) is a Borel measure on a compact
measure space, and in *VV we have Lfi(A) > 0, then o{A) ^ 0.
Using this characterization of Property M, we get immediately that:
COROLLARY 3. All selective ultrafilters have Property M.
642
MICHAEL BENEDIKT
The following proposition, implicit in [11], will be useful in the remaining results.
PROPOSITION 2. Let /u be a finitely-additive probability measure in the standard
universe V, and K be an internal set in *Vu, where U is an arbitrary ultrafilter. If
K = {Ki : i €ca)u, and if Lfi(K) > 0, then
{i :Lju(Kn*Ki)
> 0} € U.
Suppose not. Let Lfi(K) > e > 0. We can assume that for all i we have
fi(Ki) > e and Lfi(K n*Ki) = 0. We can now inductively construct an infinite set of
integers S such that ju{Kj ~ (J/esni ^0) > E/^ f° r e a c n ' e s> a n ^ this contradicts
the finite additivity of fi.
H
PROOF.
NOTES. Property M ultrafilters are not closed under products. In particular,
U x W can never have Property M if U and W are isomorphic: If / is the
isomorphism of co mapping U to W, then the *-clopen set in *T'
([/(»')] ~ [J] • i,j
£co)uxw
has Loeb measure 1/4, but no standard elements. For related results on the limits
of Property M, see the remarks at the end of this section.
Although we don't have closure under products, we can prove the following:
4. IfU and W are nonisomorphic selective ultrafilters, then the ultrafilter
U x W has Property M.
THEOREM
PROOF. By Lemma 1 it suffices to consider Lebesgue measure on the unit interval.
Letv4 = (Aij : i,j € co )uxw be an internal set of positive A measure in ric/xw ^ =
Vox w- Without loss of generality, we can assume the Aij are all compact, and all
have I measure greater than some fixed e.
We need some notation to handle the subtleties of the iterated ultraproduct
construction. Let
Vw = the NSU formed via W over the original superstructure V
Xw — the (finitely additive) measure °dw{X) in Vw
Vw,u = the NSU formed over the superstructure Vw via U
L2.w,u = the Loeb measure formed over Xw in Vw,u
LXUxw
= the Loeb measure formed over X in Vw
x v
A'i = {Aij : j £co)iv € Vw
At = (Ajk : j,k eco)uxw
€ VUxw,
where Ajk = Aik.
/
We will use the fact that Hf/xn (^'^) ^ isomorphic to n ^ d l w C ^ ^ ) ) ' ar "^ that
this induces a corresponding isomorphism J between internal sets in VUxW and a
subalgebra of the internal sets in Vw,u'- namely the map that sends a set
(Bjj :i,j £co)Uxw
to
({Bij : j <=co)w '• i &co)u.
This isomorphism has the property that LXUxw{A) = LXw,u{-^{A))- Also, JF
sends the VVxw set A to the Vw,u set {A\ : i G co)u, and the VUxW set At to the
Vw.u set du{A'j).
We consider a game whose plays are composed of players 1 and 2 playing positive
integers. At any stage of the game, either player can play. A partial play of the game
ULTRAFILTERS WHICH EXTEND MEASURES
643
will consist of a pair offinitesubsets of co, s and t, representing the moves made by
the players so far. Player 1 begins by playing an n such that LXVxW{An DA) > 0.
Player 2 can respond by playing any m > n so that
LXUxW{dUxW{Anm)C\An
DA) > 0.
More generally, if the partial play is (s, t) then Player 1 can play any i > max(?) so
that
f] dUxW{Ajk)r\
LXVXW(
Pi
AJV\A\>Q.
j<k
Player 2's response to a partial play of (s, t) can be any / > max(s) so that
LXUxW(
H
dUxW(Ajk) nf^Aj
DA j>0.
j<k
By & full play of the game, we mean an infinite sequence of legal moves by either
player. Any legal full play of the game can be described by the pair (S, T) where
S = {integers played by 1} and T = {integers played by 2}. If {S, T) is such a full
play, then the sequence of sets (Ajk)j€s, keT, j<k has thefiniteintersection property.
Therefore we have that:
CLAIM. If there is a full play of the game (S, T) with S e U and T G W then
a{A) jL 0.
PROOF. AS noted above, the sequence of sets (Ajk)j<=s, keT, j<k has the finite
intersection property, and since each Ajk is compact this implies that there is an x
common to all of them. For this x we have V/ G S,Vk e T ~ j x e Ajk- Since
H
S G U and Vy T ~ j e W, we have x G Ajk a.s. (U x W).
Our goal then, is to get a legal play with moves S and T in U and W. For any
partial play (s, t) let
G.s,t = { n : Player 1 can respond to (s, t) with n }
Hs,t = { « : Player 2 can respond to (s, t) with n } .
CLAIM. G3I, G
PROOF.
(t)
U and Hs<t G W.
If (s, t) is a partial play then we know that in VUxfV we have
LXuxw
dUxlv{Ajk)nf>\Akr)A\ >0.
D
Let
£=
p|
4xifU/i)nf|4
j<k
7
The homomorphism J of VUxw into K^t/ sends each dUxW{Ajk) and each ^
to standard sets—sets that are images of things in Vw under the map dy that goes
from Vw into Vw,u, hence each of them is represented in the ultrapower Y[u Vw
as the [/-equivalence class of a constant function. It follows that E is also standard
644
MICHAEL BENEDIKT
in VwjV, i.e., S{E) = du{C) for some C in Vw. Hence the isomorphism of
Vux w to Vw,u translates the equation {%) into Lkw,u{dv{C) n J?(A)) > 0, where
*f{A) = {A'n : n e m)u- Applying Proposition 2 in Vw,u to the measure kw
restricted to C, gives that
{i :LXWiU(du(C)ndv(A'i)f]Jr(A))>0}e
U.
Applying the inverse of the map S we get that
{/ :LXVxW{Er\Ai^A)
> 0 } e U.
Thus, Gir e U.
Now consider # v r .
Letting B = f]j€St A e , y<fc <fUx wUy*), D = f\ e .s ^* we have:
(t)
L/lj/x^(5nZ)n^)>0.
Applying Proposition 2, in VUxW this time, to (f) we get that
{(i,j)
ecoxco:
Lkuxfv(duxw(Dij)
where D = (D,y : i,j € co)Uxw
the above reduces to
r\dUxW{Au)
n B nD n A) > 0}
G 1/ x W,
But Ay = f\ev -^fc/> (regardless of what i is) so
| (/,;) £ cox a> :LXUxW(Bf]
f] dUxW{Akj)
n dUxW{A,j)
DD DA J > 0 I
t6.y
hence
i (*,./} ecu x co :LlVxW{
B (~) f]dUxW{AkJ)
nD HAj
>O\
&U X W.
But the above can only happen if
ljGco:LlUxW(Br\f]dUxW(Akj)nDr)Aj
>o\e
W.
This last set is (modulo the finite set max(r)) equal to Hsl, so we now have Hs, e
W.
H
For n e co, let
G„ = p | G„,
s,tC.n
#„ = f| Hst.
s,tQn
Then Gn eU, HnG W.
Using the fact that U is selective, we can get an S e U so that
Vw, n e S n > m =>• n € Gm.
(Consider the partition of [co]2 into P0 = {(m, n) : n e G m } and PQ, apply the fact
that selective implies Ramsey.)
Similarly, since W is selective there is T € W such that
Vn, m E T n> m ==>« € 77m.
645
ULTRAFILTERS WHICH EXTEND MEASURES
We now define two functions f ,g: co ==>• co, by
/ ( « ) = least element of S > n,
g(n) = least element of T > n.
Since U and W are nonisomorphic, we have 35" G U T' G W w i t h g " ( 5 ' ) n r ' = 0.
By intersecting, we can get 5" c S, T c T with g"(S") n f = 0. These S' and J '
have the property that:
whenever m £ S' ,n £ T', there is a y in the original T with m < j <n.
Similarly, since / is not an isomorphism between U and W, we can further shrink
S' and T to 5"' G £/ and r " e W so that g " ( r " ) n S" + 0. 5*", r " still satisfy
the property listed above, and also the symmetric one:
Vm G T", Vn € 5"', n > w =>• 3 ; e 5
such that
m < j < n.
We now complete the proof of the theorem by showing:
CLAIM.
(S", T") is a full play of the game.
PROOF. S" C S C GO C G(0,0) = set of legal responses to the empty partial
play, and min(S"') < m i n ( r " ) , so min(S"') is a legal opening move by 1 (in my
description of the game, we commanded Player 1 to move first). Now suppose that
we have (s, t) initial segments of S", T" respectively, such that (s, t) is a legitimate
play in the game.
CASE 1. The next move after (s, t) is made by Player 2. Let k be m i n ( r " ~ t).
Then k > max(s), and since k e T" and max(s) e S", there is a j £ T with
max(^) < j < k. But then
j < k are elements of T = > k e H,
=>k G H(s>t)
^ k
(since s, t c j)
is a legitimate Player 2 response to (s,t).
CASE 2. The next move after (s, t) is made by Player 1. Let n = min(5"' ~ s).
We have n > max(?), « G S', max(f) G T". Again, by the property of 5"' and T",
we have anm £ S with max(t) <m <n. Then
n£Gm=>n
G G<.Sj,) => (s U { « } , r)
is a partial play of the game.
H
We will now extend Theorem 4 to countable products.
Let (U„) i <„<eu be a sequence of pairwise nonisomorphic selective ultrafilters. We
construct a sequence of ultrafilters (W n )\< n < m as follows:
WX = Uu
THEOREM
Wn+l = t/„ +1 x JFBJ
5. W (a^ defined above) has Property M.
PF = £
f^,.
646
MICHAEL BENEDIKT
We will need an extension of Proposition 2 to ultraproducts.
Let ({Xj,I,i,jUi))ieio be finitely-additive probability spaces and U be an ultrafilter.
Suppose there exists for each i mappings 7r,;: Xt =>• Xj defined for U-almost every
j such that Ttjj preserves measure:
PROOF.
V/ < ;, VA e 2,, mU)
= yU/foyU)),
and such that the mappings commute: iijk o nij = nik.
Let U be an ultrafilter and for each i let Wt be a set such that X-, c Wt and
co <z Wt, and consider the model M = X[v{Vm{ Wt), e). In this model we have
a *-measure ju — (JUJ : / £ co)v. The hypothesis guarantees that for each fixed
i and each S £ 2, the set n(S) — (nyiS) : j £ co)u is in domain(//) and has
/it{S).
/ a-measure=
LEMMA6. Let A = (At : i £ co)u £ M be internal with A 6 domain(//) aw/
suppose we have e £ 5ft+ SMC/Z ?/;af //(^4) > e. 77!en 35 > 0 SHC/; ?/za?
{i:/iUn*U,-))><5}
w /« t/.
PROOF. Fix J = any integer > 2/e and let<5 = e/2 J. If the conclusion fails, then
we can assume that
(l)Vi fii(Ai) > e , but
(2)Vi pj{Aj mttjUi)) <5 &.&. j .
Using (2), construct a sequence {nk)k<j such that, for i <k < J,
Mnk(A„k nnni„t(Ani))
<8.
Then the sets
KJ
~U
(An,)
i<k
are J disjoint sets with JUJ -measure at least (e — J -S) > e/2, which contradicts the
finite additivity of JUJ .
H
Next, we have to think of W as being indexed by the leaves of a well-founded
tree, rather than its "official definition" as ^2V W„.
Let L„ = co", and L = { n"~a : a £ L„}. From now on, we will take L„ to be the
index set for W„, andL to be the index set for W, with S £ W if and only ifSn £ Wn
a.s. Uw where S„ = { a : n^a £ S }.
Let
T — {all initial segments of L} — { n^a : \a\ < n } .
Foranyo- £ ^et/-(a•)betherankofo•inthetree7 , : i.e.,a>if<7 — 0, car(«r) —(|o-| —1)
otherwise, where car(er) is the first element of a. Let
T" — {nodes of rank n} — {a : car(er) - (|er| - 1) = n }.
For every m < co W can be decomposed into
Wm x Wm
where Wm is an ultrafilter on Tm, the nodes of rank m, as follows:
ULTRAFILTERS WHICH EXTEND MEASURES
647
For n > m, set W^ = U„ x ([/„_, x • • • x (U m+l )). W^ is an ultrafilter on co"~m
and S £ W if and only if
{n : {a : {r : Sn~a~, £ Wm] £ Wnm}} £ [/„,
where S^- = { y : <S~y G 5 }.
S o W ^ E[/,„(^m x wm), so setting ^ m = J2Va wm' we have W ^ Wm x Wm.
Note that the index set of Wm is { {«} x a : |er| = °n - m } = T m . Set PF0 = PF.
Note that T° = L.
In the last proof we switched back and forth between two universes, Vw,u and
VuxW. In this one we will have countably many universes:
For n < co define
V„ = the NSU formed via W„.
V" = the NSU formed over V„ via
V
w
W.
= the ultraproduct of ( V„ : n £ co) via Uw.
Vo — the standard universe (formed over a base set containing X, N, and 3?).
F — the NSU formed via
over the standard universe.
We now get versions of the Loeb measure for each universe:
lo = X.
X° = the Loeb measure formed over Xo in V°.
X„ = the finitely-additive measure °dWn (X), formed in V„.
X" — the Loeb measure formed over X„ in V".
Xm = the Loeb measure formed over (X" : n G co) in Vw.
The isomorphism of W and W m x Wm and the isomorphism of W and ^ ^ W„
yield homomorphisms ip fox p < co from internal sets in V° into F ? such that
k°(A) = V{ip(A)).
Namely: /',, sends any set of the form (Ba : a G L)w to (Bs : 8 e Tp)Wp,
where B$ G F p is defined by (Bs~T : t £ Lp)wp- i<o sends (Ba : a G L ) w to
(B„ : n e co), where 5„ G F„ is defined by (B„~ T : r £ Ln)Uai.
Let (^4a)aeL be such that all A„s are compact with X{Aa) > e > 0.
We now associate with each tree node a £ T a set A" (the "<r-fiber" of A), as
follows:
Let n = r{a) > 0,
A" = (Aa~s:6eL„)w,eVn,
if<r^0.
For anyCT(including 0) with r(a) = n, and m £ co with n > m define:
Where y <= a is the sequence y' where y'\ dom(er) = a and y' = y on dom(y) ~
dom(o-).
Aa'm is well-defined, since fFm-almost-surely, a thing of rank «? has an initial
segment of rank n (all nodes of rank m with length bigger than n — m have such an
initial segment).
648
MICHAEL BENEDIKT
We also define:
A9*a = (AW
:n£co)Uw£
V'.
Note that A®'0 = A and
Axfi = dw(AT)
ifr(r) = 0.
We trace through what the mappings im do to these sets: im(Az'°) = AT,m for
m < r{t), for any T.
!„,(^4 r0 )forr(T) < m < co is the standard element of Vm dw"{nr[r)m{Ax)), where
rtij is the embedding of Vt into F,.
im{Ar'°) for T ^ 0 is a "standard" element of V<° = n(Ar) = (nmj(Az) :j Geo),
where nmj is the embedding of Vm into F ; . Finally, we have ia(A9,0) = A®<w.
We will need the following standard fact about distinct /"-point ultrafilters:
PROPOSITION 3 (Folklore). For any collection (C/,-),-ecu of distinct p-point ultrafilters, there are pairwise disjoint sets (Z),),Gcu with Dj £ [/,-.
PROOF. First, for each fixed /, get an St £ Ut ~ 1J; / • U,- as follows: for each
j / / choose a set S,-/ G £/,• ~ Uj, then take any S, £ £/,- with IS1,- ~ Sjj\ < co for
each j . Once we have each St, we then let D, — S,- ~ |J <( Sj.
H
We will also make use of the following property of nonisomorphic selective
ultrafilters due to Blass, which is a strengthening of the fact that J^D ^i ' s a Q~
point:
LEMMA 7 ([5, Corollary 7.2]). For any h: co=>co, and nonisomorphic selective
ultrafilters (Un)n€w, there are sets K„ £ U„ such that
Vx,y G ( J ^ „ [x<y=>h(x)
<y].
n
Fix a collection of pairwise disjoint sets (A)i<;<co with D, G £/,-. By thickening
these sets, we can assume that they partition co. For a £ T, write Da for Z>r(<j)For w e a) let row(m) = the r such that m £ Z),. For t £ co<w, say ? is race if ? is
increasing = (t0 < • • • < t„), t £ T,t0 £ Dw, and Vz < n ti+\ £ £><,„...,,>.
Equivalently, a nice sequence is either
(1) the empty sequence,
(2) a singleton in Dm,
(3) an increasing sequence (to < • • • < t„) with n < to, to £ Dw, row(?i) = to,
and row(f/+i) = row(/ ; ) — 1 for j > 0. This equivalence follows from the fact that
r({t0...tj+i))
=
r((to...tj))-l
for j > 0.
For s £ [co]<w let * t = the set of nice sequences from 5. Notice 0 G s | for every s.
Say that s is good if
A
°(n A a f i ) ^°Ve.vT
CLAIM.
If
S
'
is good then Mm < co 3X £ Um such that Vw £ X, s U {«} is good.
ULTRAFILTERS WHICH EXTEND MEASURES
649
PROOF.
CASE 1. m < co.
Apply / m _i to the inequality A f l e s T ^ a '°) ^
r~'(
0> I n ym l
~
we
Set
f| im_,U"-°)n f) i m -i(#°))>o.
r()')<m
y&'T
The set C\r(y)<m im-i(Ayft)
while
r(y)>m
ye.sT
is standard in F m _ 1 = rf^-i (5") for some S in F m _i,
r{y)>m
r(y)>_m
so we have
A m - i C^„-.(5)n n ^ A m - i N )>o
r(}>)>m
yen
Am ' is °<3?n/m-i (i m _i), so applying Lemma 6 with all /i, = km-\, we get
3XeWm~[
such that
WeX
yes]
1
/e.vt
m
Since W""- = W x Um, we can decompose the X £ Wm~x above to get a <r 6 T"
and P £ Um such that
r
(y)^m
yes}
Hence, for this a we have:
l
r- (d^-,(s)n
r(y)>m
yes]
f) /'"-'n f| d^-MTn1)) >o v«ei>.
r(}")>m
yes]
r{y)—m
y€s]
But if rank(y) = m — rank(er), then A„~n
that
l
= Ay~n, so we have: 3P e Um such
Am-1C</w„.-i(5)n p | Ay'm~ln fl
r(y)>m
yes]
J ^ - I U ^ " " ) ] >O
V«GP
r(y)=m
yes]
and applying the inverse of the isomorphism / m _i we get: 3 P e C/m such that
x°( f] # ° n p) ^ n p ^
r{y)<m
yes]
r{y)>m
yes]
>0 v«ei>.
r(y)=m
yes]
By intersecting with Dm we can get P c Dm, and we can arrange min(P) > max(*)
as well.
650
MICHAEL BENEDIKT
For every n G P we get
s U {«}T C s~\ U { y^n : r{y) — m }.
This is because nice sequences have to be increasing (so n > max(s) implies n can
only show up at the tail of a nice sequence), and because
y^n i s n i c e = > r ( y ) — row(n) = m.
But then we have A°(n,, e5U{ „} T ^ , 0 ) > 0 a.e. Um.
CASE 2. m = co.
Apply im to the inequality AfXe^T
y4a
'°) ^ °-
I n F£
° w e Set
^ ( fl U^'°)nU^)) >0.
The set r\r(y)<co, yesi ! ™(^ y, °) i s "standard" in Vm = n{S) for some S in some Vm
(since there are only finitely many y in sf, we can take m to be any bound on the
rank of the nonempty elements of s | ) while im(A) = (A^ : w G co) M, so we have
A o , (rf £ / „(s)ni o ,U))>0.
A" is the standard part of (A„ : « G co) [/„ • For every m < n the embedding of
the ultrapower over Wm into the ultrapower over W„ gives a measure-preserving
mapping of domain(/l m ) into domain(A„), and the mappings commute as specified
in Lemma 6 (since the corresponding embedding of ultrapowers commute), so
Lemma 6 applies with //„ = A„ to give
3X G Ua such that
n 4 , U ) n Jr(j4{"})) > 0
Vn £ X r{n{S)
and applying the inverse of the isomorphism iw we get: 3X G Um such that
A°C f| ^ , o n / 0 n ^ > o v«eP.
>(y)«u
'
By intersecting with £)„ we can get P c D m and we can also assume min(P) >
max(s) as before. Since P c Dw, for every n G P we get S U { « } | C ST U {«}. But
then we have A°(D y6jU{ „ }T ^ ' ° ) > 0 a.e. Um(n).
H
For any s, and any 1 < k < co let
Gj,jt = { n G D,t : s U {«} is good }
if * is good
G^fc = all of co, otherwise.
Then if we set Giik = f]s<zi G^, for 1 < k < co, i G co, we have that G,^ G £4.
Since £4 is selective, we can get Sk £ Uk so that Vm, n E Sk m < n ==>« G G m ^.
We now want to shrink each Sk to an S'k G f4 so that we have for every i / j <co:
(tij)
Vm e S-, n G S'j m<n =>• 3x G S/ such that
m < x < n.
Let Co be defined to be Um for the next section, and let SQ = Sw.
Let g(i) = least x > i in Srow(,), and let A(«) = max (< „ g(i).
ULTRAFILTERS WHICH EXTEND MEASURES
651
By Lemma 7 there are sets Sj for i > 0 so that
Vx,y€\Js'n[x<y^h(x)<y].
n
By intersecting we can assume S[ c St. It is clear that the Sjs satisfy (t,-,)Set S^ = Sl,
Pw=\J
S'k.
We now show that:
CLAIM. Every initial segment ofPw is good.
By definition, 0 is always good. Suppose we have already shown that s =
{so... s„} is good. We know s„+\ G S'k for some k < co. If sn G S'k for the same k,
we have
•Wi £ 5*, s„ e. Sk A s„+l > sn =>• sn+l G GJfMfc
=>• J„+I G G^fc
=^5U{j„+i}isgood.
If J„ G S< with y ' ^ A:, then
s„+i G Sj, *„ G S'k A sn+i > sn = > 3x G Sy
=>sn+i
such that
s„ < x < sn+\
G Gx j
=>Sn+\ G G^j
=>iU{5'„ + i}isgood.
H
For7<o>,letP, = s ; u | J , < , < , s ; .
CLAIM.
(1) V; > 0 j G 5^, { a G L, : . T a €/»,-?}€ **y( 2 ) P ^ | n L € fr.
PROOF.
(1) Let y G S^ and let a be any increasing sequence from Sj x ... S[. Then j"a
is nice, since it satisfies part (3) of the equivalent characterization of niceness, and
thus j^a G Pj]. Since S't G C/, for 1 < i < j , the set of increasing sequences from
S'j x • • • x S[ is in W), and (1) follows.
(2) follows from (1), since S'm e Um, and
Plv]nL = {ra:jeS^aePJ]nLj}.
H
Let P = iVT n L. We now complete the proof of Theorem 5 by showing:
CLAIM.
f)oePA°^®PROOF. It suffices to show that the collection {Aa}a€p has the finite intersection
property. Given o\, ..., a„ G P, 3 an initial segment s of Pw such that each
Oi G s\.
Ao ( n Aa)=A° (n ^u,,))=A° ( n ^°)
652
MICHAEL BENEDIKT
(since a has rank 0) and since s is good, A°((\ 6 v T Aafi) / 0, and the conclusion
follows.
H
Similar techniques can be used to prove that if (Ua)a<W] is a sequence of nonisomorphic selective ultrafilters, and we build a sequence of ultrafilters {Wa)a<wi by
setting
Wo = U0,
Wa+X = Ua+l x Wa,
Wfi — y ^ ^j»„
f° r P a limit ordinal,
where /?„ is cofinal in /?. Then each Wa has Property M.
As a corollary to the previous result, we see that a sum of nonisomorphic selective
ultrafilters has Property M.
COROLLARY 8. If {Uj)j€to is a sequence ofpairwise nonisomorphic selective ultrafilters, and D is another selective ultrafilter, then the ultrafilter W = Y^D ^i nas
Property M.
PROOF. Without loss of generality we can assume that D is not isomorphic to
any of the [/,-, since it is isomorphic to at most one, and the behavior of W is
determined by what occurs on the remaining rows. Each t/, <RK Wt, where Wt
is defined from Ut a3 in Theorem 5. Hence, letting D = Uw, we get W <RK WOJ,
where Wm — J2o Wj. By Theorem 5, Ww is Property M, but it is easy to see that
the Property M ultrafilters are downward closed under <RK (a stronger statement
is shown in Lemma 11 later on), and hence W is Property M.
H
COROLLARY 9 (MA). There is an ultrafilter that has Property M and is a Q-point,
but is not selective.
PROOF. It is known that the sum of nonisomorphic selective ultrafilters is a Qpoint and not selective [17], and the result now follows from the previous corollary.
H
Elsewhere [2] we have shown that there are no containments provable in ZFC
among Property M and boolean combination of the classes of selective, semiselective, .P-point, g-point, and Rapid ultrafilters, other than the ones that follow
from boolean algebra from selective c Property M and the containments already
known for those classes (e.g., selective c P-point). On the other hand, it is shown
in [2] that the Property M ultrafilters properly contain the Property C ultrafilters
of Daguenet [8], with containment being proper under MA. Shelah has shown that
Property M ultrafilters are all nowhere dense [19]. Shelah has recently shown that
there may be no nowhere dense ultrafilters [19], and hence we know that "There
exist Property M ultrafilters" is independent of ZFC.
§4. Applications. We now explore some consequences of Property M.
LEMMA 10. For {X,T.,fi) perfect, S an L/u-measurable set in a Property M universe,
we have
dinner
(a{S)) < LM(S) <
pouter
(Ms)).
ULTRAFILTERS WHICH EXTEND MEASURES
653
In particular,
Lfi(S) = fi((r(S))
whenever <r{S) is ju-measurable.
PROOF. We prove the second inequality first. Suppose that L/i(S) > jUouter(c(S))Then there is a //-measurable A D <r{S) such that LI{A) < Lju(S). Then S ~ *A is
L//-measurable with positive Loeb measure, and a(S ~ *A) = a{S) ~ a{*A) — 0.
This contradicts Property M. The first inequality follows from applying the second
XoSc.
H
The primary motivation for studying Property M is its application to the extension
of perfect measures. In Property M universes, we have the ability to push down the
Loeb measure onto the original sample space.
For any perfect measure (X,l,,/u), define the extension measure of // formed via
U', jxy (or jx when U is understood) by
domain(//) — { o(A) : A e*V and A is L/x-measurable },
fi(*U)) = LM(A).
PROPOSITION A.IfU has Property M, and jx is formed as above then:
• ji is a complete probability measure which extends //. Iffi is nonatomic, then ji is
a strict extension of fi {i.e., domain(//) strictly contains S).
• For any sequence of ju-measurable sets (Ai)i£c}, if A = U-limAj, we have
fi{A) = U-\im ju(Aj), and if fi{Ai) > e > 0 we have also
3X e U\/i EX fi(ADAi)
>0.
PROOF. First note that that if 3Z,/*-measurable A, B with a(A) = o{B), then
a {A A B) = 0. Therefore, by Corollary 10, Ljx{A A B) = 0 and hence Lju(A) =
Lfi(B). So fi is well-defined. If A e l , then A = a{*A), and it follows from this
that jx extends ii. The completeness of// follows from the completeness of the Loeb
measure.
If// is nonatomic then there is a sequence of sets {Aj)ieaj such that Vi fi{Ai) = e
with 0 < e < 1, and the At pairwise independent.
Take K € *N ~ N and let D = a{AK). Then D is //-measurable, and we show
D $. S. To see this, note that by Transfer we have AK independent of each A,.
Hence Lju{AK ~ A{) > 0, and therefore there is some standard positive real e such
thatVz' G co /i(D ~ At) > 0. IfD e S we then have LL(D ~ At) > e, since// extends
fi. By transfer again, we get Lfi(D ~ AK) > 0, but this contradicts Property M,
since a{D ~ AK) — 0.
The rest of the proposition follows from the definitions and from Proposition 2.
H
The measure jx shares many properties with the Loeb measure L/x. In particular,
extension measures can be used to obtain saturated adapted spaces (see [12], [13])
and to construct Anderson's Brownian Motion [1].
In addition, extension measures are comparable to the original space in ways not
applicable to Loeb measures:
PROPOSITION 5. If (X, Z, //) and {Y, F, v) are perfect, then:
654
MICHAEL BENEDIKT
(1) If F: X => Y is (ju,v)-measurable {S £ T implies F~S{S) £ E), then F is
(fi, v) -measurable.
(2) If F: X => Y is (JU,V) inverse-measure-preserving: i.e., fi{F~](S)) = v(S)
VS e r , then F is (fi, v)-inverse-measure-preserving.
(3) /( « v implies / l < v .
PROOF.
(1) follows from the fact that VX e T , a(F~l{X)) = F-l(a(X)),
and (2) from
the fact that F preserves measure implies that *F preserves *-measure.
If [i -C v then
Ve> 0 3d > 0
such that
v(S) < S =4> ju{S) < e.
From this we see that Lv{S) = 0 implies L/u(S) = 0, for any S in *V. It then
H
follows that p. <€.v.
Extension measures may also be compared to one another (for different Property M ultrafilters), as well as to the original measure space. In particular, we can
ask:
(1) For two Property M ultrafilters U and W, and a fixed perfect measure
(X,l.,fi}, U and W both give extensions fiu and fiw of//. Unlike the two Loeb
measures Lv/i and LWju, these two measures both live on the same set X, so it
is possible to ask: are fiu and ftw the same measure, and if they are different, do
they agree on the intersection of their domains? Are there sets that are universally
measurable for all extension measures of /u over all Property M ultrafilters, other
than the sets in S?
(2) It is easy to show that fiu has no extension measure via U if ju is atomless
(and hence fiu is almost never perfect, as opposed to Lufi, which is always perfect).
Thus, there is no way of iterating the extension process using the same ultrafilter U
each time. But can we extend //[/by using a different Property M ultrafilter W1
We can answer these questions in the special case where the ultrafilters are selective, using the results of the previous section.
We start by settling question (1) in the case where U and W are selective ultrafilters. When U and W are isomorphic, then the situation is clear:
PROPOSITION
6. IfU=
W, then the measures fiu and fiw are the same.
PROOF. Let / be a permutation of co such that f"{W)
= U. Let A = a{A) for
some set A internal in the NSU *VV formed via U.
Then A = U-limAj, where A = (At : i € co)u, and fiu(A) = U-\imfi(Aj).
But U-limAi = W-HmAf^, and so A = a{B), where B = {Af^ : i e co)w
This means that A = domam{fiw) and fiw{A) =
W-lim/xiAf^).
But W-limju(Af^)
— U-limfi(Ai), so we have that fiw and fiu agree on sets
of the form
{ o{A) : A internal in *VV } = {<?{A) : A internal in *VW }.
Using the fact that the measures are both inner and outer regular with respect to
sets of the above form, we get that they must have the same domain, and that they
agree on their common domain.
H
LEMMA 11. If U <RK W and W has Property M then:
ULTRAFILTERS WHICH EXTEND MEASURES
655
(a) U has Property M.
(b) For any perfect measure (X, S, fi), jxw extends fiuPROOF.
Suppose (Ai)iew
are measurable sets with fi(Aj) > e > 0. Then
U-limAi = W-lim A f(j),
where / is such that U = f"( W). Since W has Property M,
W- lim Af{i)
^ 0 ==> U- lim A,• ^ 0 ==• £/ has Property M.
Since £/ has Property M, we know that juu exists and that juu(U-lim A,) —
U-limfi(Ai). But
jiw{U-limAt)
= {iw(W-lim
A f{i)) = f F - l i m / i U / ( i ) )
= £/-lim//(/!/) = / / [ / ( f / - l i m ^ / ) .
So jxw gives the same value as fiv does on all sets in domain(//j/) of the form
U- lim /4,-, y4(- e l . But domain(/i ^) is completion of the <r-algebra generated by sets
of this form (since the Loeb measure is the completion of the <7-algebra generated by
the internal sets), so we can see that ju w agrees with ju v on all of domain(//if). H
THEOREM 12. If U and W are nonisomorphic selective ultrafilters, and (X, S, /u) is
any atomless perfect probability space, then domain(/in/) ^ domain(/i[/), but feu
and few still agree on the intersection of their domains.
PROOF. U X W has Property M, by Theorem 4, and therefore, by Lemma 11 fiu
and (i w have a common extension, fiuxw- It follows immediately that ft v and fi. w
agree on the intersection of their domains. Let (v4,),eco be sets with n{At) = 1/2
that are independent. We will show that (Aj : i e co)u £ domain(fiw)- This will
follow quickly from the following:
CLAIM. If U and W are as above, and S 6 domain(/i v) n d o m a i n ^ w), then there
is a fi-measurable K such that /iu(K AS) = 0 and jx w (K A S) = 0.
PROOF. If S is in both domains, then there are internal sets C and (iV,), em in *VV
such that Ljuu{Nj) =>• 0, and S Aa(C) C a(f]iemNi).
Similarly, there are internal
D a n d P , i n * F ^ s u c h t h a t L / i ^ ( P ; ) : = > O a n d S A < T ( D ) co-(f|,.P,). Considerthe
two "diagonal" maps
Fu: *Vu=^*VWxu
that sends (Aj : j e co)u to (Aj : i,j
Gco)ivxu,
FW'- *Vw =^*Vw*u
that sends (At : i e co)w to (At : i,j
£co)WxU.
and
Both Fv and Fw respect finite boolean operations, and
Luii{A) = Lwxul*{Fu{A)),
LWfi{A)
In addition, we have a {A) = o(Fu(A)), and a{B) =
Now we have the following chain of inclusions:
a{C)Aa{D)ccj(C}N^\
Ua(C]P^
=
LWxUfi{Fw(A)).
a(Fw{B)).
C f\o{Fv{Nt))
U
f)a{Fw{P,))
656
MICHAEL BENEDIKT
and therefore:
a{Fv{C))
Aa{Fw{D))
C f]a{Fu{Ni))
U
f]a{Flv{Pl)),
i
i
which gives
(3)
a{Fu{C) AFW{D)) d
a(f]Fu{Ni)\yjP\a{Fw{Pi)).
^
i
'
i
7
(3) implies that the *PVX v set T ^ (C) A i<V (Z>) must have Loeb measure 0: If this
were not the case, setting
E =
we have LWxul*{E)
f]{Fu{Ni))uf]{Fw{Pi)),
= 0, since E is the union of two Loeb-null sets, and this gives:
LWxUli{Fv{C)AFw{D)
~E)> 0=^a(Fu(C)AFw(D)
~ E) ^ 0,
(since W x U has Property M) and this contradicts (3).
SoLWxUju(Fu(C)AFiv(D)) = 0. Letting C = {Q)vandD = (A) w we have
\fk3S GW xU V{i,j) e S //(Cy A A) < p
which implies
VA: 3ak 3X € W V/ € X p(CBk
ADj)<
1
•V* Bajt L ^ ( 4 ( C „ t ) A D ) <
1
T.
jfc'
Choose an ak for each k satisfying this. For j < k £ a>
n{Ca]ACak)<LWfi{dw{Cai)AD)
+
Lwn{dw{Cak)AD)<
2
so the characteristic functions of the Cajs are Cauchy in measure, hence converge in
measure, hence have a subsequence that converges a.s. /u. The limit function of this
subsequence is clearly {0, l}-valued, and thus can be taken to be the characteristic
function of some set K. For this K we have:
Lwfi{dw(K)AD)
= 0=^juw{KA(j{D)) = 0^/iw{KAS)
=0.
But also:
Lwn{dw{K) AD) = 0=^LWxuM{diVxU{K)
=^LWxU/u{dWxU{K)
=^LuM(du(K)
So the Claim is true.
AFW{D)) = 0
AFV{C)) = 0
AC) = 0^fiu(KAa{C))
=0
H
We now return to the proof of Theorem 12: If A = (A, : i e w)v £ domam{jUw)
then there is a //-measurable K such that fiu{AAK)
= fiw {AAK) = 0. But since
ULTRAFILTERS WHICH EXTEND MEASURES
657
the collection {Aj)ieu) is independent, and each At has measure 1/2, we get
lim fi(Ai n L) = - • M(L)
/—»oo
2
VX G Z,
hence by Proposition 5 we have
ftu{AnK)=
U-\imfiu{AinK)
=
yju(K),
and this clearly cannot be, since juu(A) — 1/2 and fiu{A A K) = 0.
H
7
If U and W are nonisomorphic selective ultrafilters and (X,'L,n) is perfect, it is
not in general the case that
domain(//£/) n domain(/i^) = Z.
For example, if v is the usual measure on £P(co), and {C/,}, etu , {Wj}ieco
ultrafilters isomorphic to U and W, respectively. Then
are distinct
C\ Ui n W7, G domain(/2(y) ndomain(/}n/),
but is not //-measurable (since the intersection of co many ultrafilters is never //measurable [21]). However, we do have:
PROPOSITION 7 (MA). If (X, Z, fi) is perfect and countably generated, then
C\
domain(/i[/) — Z.
V selective
We will show that any D £ Z cannot be in the intersection.
Fix D £ Z, and an F c Z of size c such that every measurable set of positive
measure contains a positive measure set in F.
There are D\, Di G Z such that D\ <z D <z D2 and /z;nner(-Di) = pinner (-D).
,«outer(£>2) = /"outer(i>)- Let C = D2 ~ I>i. Then //(C) > 0 and both D and £»c
have full outer measure in C.
Let {Sa)a<c enumerate all countable sequences (Sf}i€a> such that
PROOF.
Vi (S? G F A Sf C C) A 3e > 0 Vi fi(S?) > e.
Let (fa)a<c list all partitions of co. We construct a sequence of sets
satisfying Va:
(a)
p<a=>\Xa~Xp\<a>.
(b)
(c)
(Xa)a<c
f|
Either
S? n D / 0 A p | S° n Z)c / 0.
3i fa(i)
D Xa+l
or
Vz \Xa+l n / a ( / ) | < 1.
At limit ordinals a we get Xa with |X a ~ A^| < co using MA.
At stage a + 1, fix E such that Vz /i(S?) > s. For each z e Xa let G, = Sf x S?,
and G = ( C n Z ) ) x ( C n D c ) . G has full outer measure in the measure space
(C x C,Z|C x Z|C,// x //), so G must hit the set H = O . (J(. . G,, since this set
has measure > £2. Let (x, y) £ G nH, and set
Xa+1={zGZQ:{x,j}cS,a}.
Then Xa+\ satisfies (a) and (b). It is easy to shrink Xa+\ so that it also satisfies (c).
658
MICHAEL BENEDIKT
(c) and (a) guarantee that {Xa}a<c generate an ultrafilter U, and also that U is
selective. If D e domain(//[/), then either fiu(D n C) > 0 or fiv(Dc n C) > 0.
Suppose, without loss of generality, that /if/ (Z> flC) > 0. Then there is a sequence
{Si)jem with ^(S1,) bounded away from 0 and a{( St : i e co)y) c D DC. We can
assume St € F, and 5,- c C. But this is impossible, since the construction of U
H
assures that for any such (S,-) there is an x £ <r(( Sf : / e co )y) n Z) r .
Shelah [20] has shown that there may be only one selective ultranlter, in which
case the intersection of all the extensions by selective ultrafilters is clearly not equal
to the original S-algebra.
THEOREM 13. For any perfect measure n there is a measure /u^, which extends {iu
for every selective ultrafilter U.
PROOF. For any countable collection of selective ultrafilters (t/,-),-ecu Theorem 5
gives a measure simultaneously extending each juVi: form the ultranlter W in
Theorem 5. This ultranlter has Property M by Theorem 5, and the corresponding
extension measure fiw extends each jxUr Since any countable subcollection of
{ fiy : U selective } has an extension, there must be one extension of all of them. H
For A, B c 8P(co), the Wadge Ordering <w is defined by: A <w B if there is a
continuous F: &>(co) -> &>(CD) such that F~]{B) = A.
Let v be the usual measure on 9*(co). For ultrafilters U and W, say U <v W if
and only if 3 a v-measurable F: 9"(co) —• <P(co) such that F~1{W) = U.
<v is not transitive (since measurable functions are not closed under composition), however, < v is a weakening of the Wadge ordering as well as the Rudin-Keisler
ordering on ultrafilters, and so can be used to gain information about these orderings.
In [4], Blass proved that if U and W are selective, then U <w W if and only if
U <RK W. We show that the following holds:
COROLLARY 14. If U and W are selective ultrafilters, then U < v W if and only if
U <RK W.
PROOF. It is clear that U <RK W implies U <v W. It suffices then to show that
if U and W are nonisomorphic, then ->(U <v W).
If U <v W then fix an F witnessing this, and let 5, = F _ 1 ([/]). Then we
have U = a({Bi : i € ca)w), so in particular, U e domain(/ln/). However
U = ([i]'. i € co)u, and the sets {[i]},-e(u are independent with measure 1/2, and
the proof of Theorem 12 shows then, that a({ [i] : i € co )a) £ domain(//iy).
H
15. IfUis selective and W = J2u ^< for nonprincipal ultrafilters Wt,
then ->(W < v U), and hence ->(W <w U).
THEOREM
PROOF. Suppose not. By considering W as an ultrafilter on co instead of co x co,
we get that there is a partition of co into sets {R{i)}iem and ultrafilters Wt on co
with R(i) e Wi such that W = t/-lim Wt. As in Corollary 14, W <v U implies
there is J? € *VV with a{B) = W.
For each i and each S e *^(Uy>,
R(j))let
A\s=lxe*&>(\J*R{j)\
:S\JxeB\.
ULTRAFILTERS WHICH EXTEND MEASURES
659
Let v, and v' be the usual measures on £P(\J •<,- R(j)) and &>({] •>,- R{j)), respectively, and
| 5 G W | J / ? ( / ) ) : i i / v / M i ) = OorLuv.-Uj
The Keisler-Fubini Theorem (see [1]) implies that ^4, is Lvv'-measurable
for Ljyv' almost every S, ^4^ is Lv,-measurable.
For any S c Uy>, &U), either
Vxc[jR(j)*SU*
x£B
or
Vx c [J R(j) *S U* x <£ B,
depending on whether S is in W or not.
So since U has Property M, for any S c |J/>i ^ 0 ' )
measurable, we must have
LuVi(Ais)
So by Property M again, Luv'(Ai)
Let
and that
sucn tnat
^ ' s *s ^ f v i -
= 0 o r 1.
= 1.
B, = {Se M ( J *(;)) : I^v,-(^) = 1 }.
j>>
Then B is within a i y v null set of B,: ®* &(|J;<,
R{j)), where
5"i <g> S2 = { sl U s2 : *i G 5i A 52 € 5 2 }.
Take an internal set C, such that Lf/v'(C, A 2?,) — 0, and let D, = Q ®*
•^(U,<,-*(;)).
Then each Z>, is internal and Vz, 5 is within a Lvv null set of A . By saturation,
there is a A: £* N ~ W and sets Doo e* ^(co), C ^ e* &{{jJ>K R{J)) so that
Doo = Coo0&(
(J /?(/)Y
Coo is Lc/v^ measurable,
Doo is within an Luv null of each Z),
(and hence within a null set of 5 ) .
Since every LvvK -measurable set is within a LuvK null set of a *-clopen set in
&{{}j>K R(J)), we can take C ^ to be *-clopen.
Let Coo = ( Coo(«) : n e co)u and A- = (Kn : n e co)u- Each COO(H) isaclopen
subset of 3°(Uj>Kn R{j)), so Coo(«) is a finite boolean combination of finitely many
subbasic clopen sets [/] for i e U/>JC„ R(j)- Let support(Coo(«)) mean
{ i : [z] is mentioned in Coo(n) }.
Then support(Coo(w)) is a finite subset of (J
Let
>KII
R(j).
Z>,°° = max{ m : JR(wi) n support(Coo(0) ^ 0 } + 1.
660
MICHAEL BENEDIKT
Since U is selective, we can get S G U such that
Vi,y €Si<j=>Df°
<jAi<
Kj.
2
(Consider the partition of [co] into { {i,j} : D°° < jAi < Kj } and its complement,
and use that K is infinite.) So:
Vi,j eSi<j=^R(j)nsupport(C00(i))
= 0,
since Df° < j , and
Vi,j eSi<j=>
R{i) n supportCCooO")) = 0,
since i < Kj s u p p o r t ^ (;)) C [jm>K. R(m).
For every / e S, let /,-: &>{R{i)) =$-&>{R{i)) be defined by
ft{T)
= ( r nsupportCCooO'))) U (R{i) n r
~ support(C 0o (i)),
for T c /?(0- / i doesn't affect T on support(C 00 (f)), and complements T off of
support(Coo(0).
Since / , is the product of the identity map on support(C oo (0) n R(i) and the
complementation map on R(i) ~ support(C o o (0), / / is measure-preserving for
the usual measure on &(R(i)). In addition:
(1) / , is a bijection, and /,- =
ffl.
(2) Vw e S, MX c i?(0, V« e support(C 00 (w)), n e l if and only if n € /,-(A r ).
This is true for m = i, by the definition of / , , and for m ^ i by the fact that
support(Coo(m)) is disjoint from R(i).
(3) /,• maps Wt n ^(R(i)) onto W? n &>(R(i)) and vice versa, since J?(z) e W,-,
fF} is an ultrafilter, and /,- is (modulo a finite set) relative complementation in R(i).
For / £ S, let / , : &>{R{i)) =>&>{R{i)) be the identity map.
Finally, let / be the "product" of the /,-s: / : &(co) = > &>{co),
f(x) = \Jfi(xnR(i)).
i
Then / is measure-preserving and / =
If Yt e Doo(i) then we have
Y, n \J *C/) e Coo(i) ^
f~x.
/(V, n (J /?(;)) e <:«,(/).
Therefore f{Yt) n U,>JC, *(./) € Q o ( 0 , since / ( F , ) n U ;>JC , *(j) and /(F,- n
U>/f, RU)) a S r e e o n support(C 00 (/)), and thus f{Yt) e Ax>(')So * / maps .Doo to Doc, and (3) implies that / maps Wt to Wf, so / maps W
to ^ ° .
By Proposition 5 / is also measure-preserving for vv.
We have that
Lvv(B
A D ^ ) = 0 => vv{a{B) A *(/>«,)) = 0
=3.vu(WAa(Doc))=0
=>vu(f(W)Af(cr(Dao)))=0
^vu(WcAa(Doc))=0,
but this is impossible, since W U Wc are disjoint sets covering & {co).
H
661
ULTRAF1LTERS WHICH EXTEND MEASURES
As a last application of the theorems of §3, we prove the result of Fremlin
mentioned in the introduction.
THEOREM 16 ([9]). For each e and every sequence of X-measurable sets {^/y);<yeco
with A(Aij) > E, we have:
X{ x : 3S, T £ [co]w such that Vi £ S,j £ T i < j => x £ Ai} } > s.
We give an alternate version of Fremlin's proof, using the results stated earlier:
PROOF. Let
J = {x :3S, T £ [cof such that Vi £S,j
£Ti<j=^x
£AU}.
Take nonisomorphic selective ultrafilters U and W, and form XUxW. The set
A = (Aij : i < j E co)uxw has £A-measure at least e, hence XUxW(cr(A)) > e.
a (A) c / : to see this, let x e a {A), and let P = { (i,j) : x e Atj }. To show
x e J we have to verify that there are sets S, T e [ca]m with
(*S, T)
{i<j:ieS!LndjeT}cP.
1
Find So £ U and J ,- e fF such that U e s o W x Ti c p- c h o o s e («.)<e» and
(bj)ieai inductively so that
• a, is an element of So bigger than all a 7 s and &,s with j < i,
• bj is an element of p| .<(. r a y .
Then for S = {a,-},-€a), r = {Z>,}/6cu, (*5, T) holds implies that a {A) C 7. So
^u*w{J) > £• But / is J2i> hence A-measurable; so, since X extends A, A(7) > s.
Now, we have proved that X{J) > e assuming the existence of two selective
ultrafilters. To see that this is true in ZF, note that Theorem 16 is equivalent to:
For every e € 5t, for every n € co, for every sequence of compact sets {Aij)i<jew
with 1(^,7) > e, and every open set O with A(O) < e - 1/n there are S, T £ [co]m
such that
f){Au~0:i£S,
j£T,
/<7}/0.
This is a Sj sentence, and hence follows from ZF if it follows from ZF + V — L
(see [15, p. 526]). Since V = L implies the existence of many selective ultrafilters,
we are done.
H
§5. Acknowledgments. These results are from the author's Ph.D. thesis, written
under the direction of Professor H. J. Keisler, to whom the author is grateful.
We also thank A. Blass, whose paper [5] led to substantial simplifications of the
proofs, P. Loeb, who suggested several of the questions in §4, D. H. Fremlin, for
giving access to his manuscript [10], and the referee, for several corrections and
simplifications.
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662
MICHAEL BENEDIKT
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