MARTINS AXIOM AND MEDIAL FUNCTIONS

No 8 - May 22

MARTINS AXIOI'1 ANTI MEDIAL FUNCTIONS

by

Dag Normann Oslo

1974

PREPRINT SERIES - Matematisk institutt, Universitetet i Oslo

In (1) Mokobodzki used the continuum hypothesis to prove a theorem on medial functions. In this note we weaken his hypothesis to Martins Axiom. We will explain both Mokobodzki's result and the contents of Martins Axiom to the reader,since he may not be familiar with both functional analysis and logic.

The references in functional analysis will be either to the book of Alfsen (2) or to the paper in which Mokobodski presented his result (1).

I will thank Erik Alfsen who introduced me to this problem and my wife Svanhild Normann who explained some of the lemmas from functional analysis to me.

2. Martins Axiom.

The original intention with Martins Axiom was not to add a "new natural axiomn to set theory, but to formulate a strong property which contained much of the information of the continuum hypothesis, but which is consistent with its negation. Thus if a statement

1'

follows by lVfartins Axiom, we may conclude:

For any n , 2

>-(o = }'(~

^{and Cf"} may both hold.

Here

>fo ,

r-

2 · denotes the cardinality of the continuum, ^;~n the n'th uncountable cardinal. The continuum hypothesis says

To formulate the axiom we'll need the following Definition:

Let P be a partially ordered set. Let p,q denote elements

of IP. D. r::.. IP is called dense, or cofinal, if

\fp ^G IP :j q ^t: fl. ( q ~ p ) •

Lemma 1:

Let { 6i} i 6 IN .; IP be a countable family of dense sets.

Then there exists an ideal G

s

P such that

~ i (:': lN ( G n !~.i -/:. 0).

Proof: Let p1 ^<.=:. 61· There is p2 ^~ p1

,

_p2 & ~. 2 and so on.

Let p E G <='> ^{( :I i)( p L - p.} _~ ⁾

.

^{Then G is an ideal.}

In the terminology of logic, G is often called generic with respect to the family _i If we assume the continuum

c.;, IN'

hypothesis, we can reformulate our hypothesis in the following way:

Statement A : Let rP be a partially ordered set, let

be a family of dense subsets of JP having cardinality less than the continuum ( i.e. countable ). Then there exists a generic ideal with respect to the family.

If we try to use this statement as a general axiom of set theory, we'll soon find out that the continuum hypothesis has to hold as well. In fact, let us prove that

A --> .CH.

Proof: Let X be an infinite set of cardinality ^L. 2

/YQ

• Let p ^E-IP if p is a finite 1-1-function defined on a finite subset of X and with values in IN. We give ~ a partial ordering by the following : p ^~ q if q is an extension of p.

Let x ^6: X • .Ux

= tP

^s tP ; p is defined on x ,} • It is not hard to see that Jjx is dense. A generic ideal with respect to

{ .6x

£

x c-X will then be a 1-1-function defined on X and with values in ®. Thus X is countable.

In Martins Axiom we have the same formulation as in A above, but we don't assume it to hold for all partial orderings.

To state it we need the following Definition:

Let IP be a partially ordered set and let { pi} i ₆ I be an indexed set from lP. We say that {Pi,~ i e I is an anti chain if for any i

I=

J, i, J c I, there is no q ^r::- JP

such that q - pi an q - pJ ^{-'>. d :::lo. ( ~.e. •} no common sucessor ⁾ We say that a partially ordered set IP satisfies the

countable antichain condition if all antichains in ~ are at most countable.

Martins Axiom, MA, then says:

i) ii) iii)

Let IP be a partially ordered set satisfying the countable antichain condition. Let { 6. i} i ₆ I be an indexed set of dense subsets of P such that the cardinality of I is less than that of the continuum, Then there exists a subset G .S: P such that

~ pgq ^f G :3rr::G~ r ^~ p, r ^~ q )

.

)lptiG '\/q ~ p ( q ''= G ⁾ 1/i<SI ⁽ G (l ~ ) _{~ 0}

In (3) this is proved to be consistent with ;{n for all n <e IN.

Note that the proof of A ==> CH does not work with A replaced by MA, since if X is uncountable, then the ^~ in that proof will not satisfy the countable antichain condition. (Look at the set of functions Px : x --:..- 0 • This forms an uncountable anti chain. )

vfe give an application, proved in the paper of Martin and

Solovay (4), to illustrate how the axiom may be used. Besides, we'll need both the lemma and the theorem later anyhow.

Lemma 2:

Let P be a topological measure space with a countable base of open sets. Let ₁^~ denote the measure, and let

f

>

0 • Let { Oi ] i c-· I be a family of open sets satis- fying

i)

V iG I

ii) i

1:

J ;>

/J.(

01 U OJ ) ^{~ E_}

Then I is at most countable.

Proof: To get a contradiction we assume that I is uncountable.

Then there is a d- , 0 < cF ^<( E: , such that

{ i ~I ; ,fot(Oi) <

cJ}

is uncountable. ( In

= {

i <:,I ; _.M-(Oi) .f:

f_- 1/n ~. Then as I = l) In, at least one In is uncountable.) Let J

=

^~ i ^€' I ; ;U ( 0 i ) <. v~·

} •

For each Oj, j ~J, let Bj be a finite union of elements f'rom the countable base such that

a) b)

Bj s oj

ft( 0, "-B.)

< (

~- cT)/2

! J J

Now.let i,j e- J be given and assume B. = B. =B.

1 J

Then ₁v< ( 0 i U 0 j ) !:: /t) ( 0 i ,, B ) + )A ( 0 j "-B ) +

ft

B <

t"-u

£.-c/

L.._ 2 ⁺ 2 ^+or= f._

But then

o. =

^{Q .•}

1 J

But there are only countably many finite subsets of a countable set, so there are at most countably many BJ's. This gives the contradiction.

Theorem 3 ( Martin - Solovay ( 4-) ) :

Assume MA, and let P and

;--<

be as above. Let { Ai

1

1 ^E- I be a family of subsets of P of ~~-measure

o.

Assume that

t\'"'o

the cardinality of I

<

^{2 • Then}

,f<( U A. )

=

0 • iEI ¹

Proof: Let ^,:E·

>

⁰ be given. Define IP by 0 t:· IP if 0 is an open set and /u.(o) < f.' • \'le order IP by inclusion. Then, by lemma 2,

IP satisfies the countable antichain condition.

Let ,6 i

=

^{f.. 0 E IP ; Ai .S 0} _•

We'll prove that ^~i is dense in IP.

Let 0 E lP. Then jv((O)

<

E.- 1/n for some n. Let 0' be open such that ^A. c 0' ,

1. ;'LA (0') <.. 1/n.

Let 01 = 0 ^i.) 0' • Then 01 ^E- D i and 0 ~ 01 •

By MA there exists a generic ideal G with respect to Let ⁰ = l.J

t

0' ; 0 ' t:- G

j .

Claim 1. ,.J!'i (0) :f E

{.6 i~

1. (::-^{· I .}

Proof: Let

r

be the set of open base elements that are included in some 0¹ G G. Then 0 = U 1-, •

If /fi.<(O)

> [

then _/{(01

uo

₂ ^{l .. : • • • •} u On )

>

^E for a finite set

o

1 , ••••• , On From 1¹ • These will on the other hand be included in some 0' G G since G is an ideal.

Claim 2: l_j Ai -~ 0 • i E-I

Proof: Let i ⁶ I. Let 0' <O; G 11 ..6 i' i.e. Ai c 0' s 0 • This proves the theorem.

3. On Mokobodzki's result.

Let K = [-1,1J IN - the Hilbert cube.

Then K is a compact convex subset of the vector space rR 1N =the set of all sequences from JR.

Addition and multiplication by a scalar are defined to be the pointwise operations.

Let f i be the i'th proJection, i.e.

f _{2• (} ~ _~ ^X n.)n-" ^{(' .} IN )

=

^{X· 1. •}

The fi's are linear, continuous functions.

Problem· 1: Does there exist a function ^{F :} K ~ R satisfying

i) lim inf { fi (x) ; i E IN} :f F(x) ^!f lim sup { fi (x) ; i c 1N

J

ii) F is universally measurable.

iii) F is linear.

Problem 2 : Can ii) in problem ¹ be sharpened to ii') F is Borel ?

Mokobodzki proves in (1) that if the continuum hypothesis holds then problem 1 has a positive solution.

We prove that his proof can be adJusted to Martins Axiom, i.e. MA ::::;:> positive solution of problem 1.

Whether problem 2 can be solved we don't know, but we prove:

If problem 2 has positive solution from CH, then problem 2 has positive solution without any set-theoretic assumptions.

We start by repeating some analysis.

Definitions:

A fUnction f : ^K -~ IR is said to be concave (convex) if:

lfx,y e:- K, ;-. c [0,1).

f( ;\x + ( 1- A)y) ~ (:') ';\ f(x) + ( 1- ~) f(y)

A function f is called upper semi-continuous if ~~ e IR

[ x ^E: K; f(x) '> ··::><.. ~ is open.

The set of upper semi-continuous functions is closed under arbitrary infimums.

Denote by S the set of all upper semi-continuous functions on K.

Lemma 4 ( Mokobodzki (1)) :

If f is concave, g,h ^E S and g 6' f , h ^L f , then there is a function f₀ ^G S such that

The proof uses Hahn-Banach, and can be found in (1) or (2).

Definition:

A net is an indexed set where the index-set I is a partial ordering satisfying

if a,b ^E- I,

3

c E: I, a ~ c

,

^{. .c} 0 - c •

A net of functions -i f i ~ i ^t:·· ₁ is increasing if i :: j f. ~f • •

~ J

Remark: If : f. _{l ~) ~} ? . _{G 1} is an increasing net of concave functions, then f(x) =sup fi(x) is concave

itS!

Let us denote by ~ the limits of increasing bounded nets of concave functions where the set of indices has cardinality less than the continuum.

By the remark all functions in

I

are concave.

If we use MA and theorem 3, we see that all functions in

~ are universally measurable.

LemmaS:

Proof:

)'(>..

Let ⁰⁽

<

2

°

be an ordinal. Let 1 ⁽ f (3 ^'!

r (;

^{< Q( be an}

increasing sequence from

2 .

Then f

=

sup f ^{r:, E. ~} f is concave by the remark. Let be a net

,..,

1:' 2

g. • By lemma 4 there

~2

is a function g such that gi ~ g ~ f ' i = 1,2.

We do this for all pairs, and repeat the operation infinitely many times ( order type IN

=

^~

).

At the end we obtain a net of functions. We may here use functions with natural ordering as indices. By a cardinality argument we see that the new net has cardinality less than the continuum.

t) ~ {. 1\1/ •

Now, let

9/, - F

^{E L. ,} <.v ==

Let {fiJ ic:I /

cp,

^{{g1 } iE I}

^{tf rf ,}

where fi,gi s

s.

(We may assume that the set of indices are the same, else we could index both by the product of the nets.)

Lemma 6:

Let /u be a measure. Assume :t-1A. Then there are two functions

(pI

^and

"'·r

^I with

(r , -

""'~'' E

L

((= C{J' =

^{4' • ~ ·"'f'}

and

qY = ·t '

almost everywhere ( /t,(. ) •

such that

Proof: Let I be the set of indices, and let A denote the set of affine functions. Let

If.l =

_l. 5 _L a ^e: A; f_{2• -} 1/n ~ a ::· g. _{J_} + 1/n • By an argument from Mokobodzki (1) we know that {~JiG I,n ~~IN has the finite intersection property, and by another argument from (1) there is a continuous function f satisfying

\1 n r;- IN 3 a ^E-

If.l ( (

I f - a I d;.'- ) < £.

l. . ./

We order I X IN by ^K

<'

^{i ,n}

'>

~

<

^{j ,m) <~> i £I J & n '!£ m •}

Then we are going to define a new net of functions a.t; r:..

If.l

such that for fixed n

l. l.

,M( If - inf

a!"f I )

^{.£ 2-n}

. I ~

~6-

,

^{;.} ( I f - sup}

^a!"f

^{1 ) ~ 2-n}

if;;I ~

If I is countable, this gives no problem. In the uncount- able case we have to use stronger principles.

I

Let n be fixed and let !P consist of all finite sets p of

.... ^,

ak such that affine functions a 1,

J (

( _max(f,a1,

. . . ^,

ak) - min(f,a1, ••• , ak)) ^d;U < 2-n

K

We order !P by inclusion.

Let 6 i

= l

^{p e. IP ; p rl ~ -/:. 0}

^{l .}

We claim that dense.

Proof: Let p ~ tP. Then ,_( ( max(f, a 1, • • • K

for some IDE IN.

By lemma 5, ^{let a}

f~

ft(

^/f a/ ⁾

be such

L.. 1/m.

Extend p by a, and we are still

that

inside IP.

6i

is

Claim 2: /P satisfies the countable antichain condition.

Proof: We reduce this problem to lemma 2.

,. 1

Let

l

^{Pt ~)} t "'" T be an indexed anti chain from \P. For each p E 1P let

oP is 0 _p

=

open

I<

K

{ (x, r

> ; (

min(f ,p) )(x)

<

^{r <.} (max(f ,p) )(x) and

max ( f, p ) - min ( f, p) ) d ;!A

= (

;vt x m )( Op) where m denotes the Lebesgue measure.

If t -/:. s f: T, let q

=

^Pt L/ Ps· Then q

t

P since { Pt

1

^{t €} T is an anti chain. Define 0 _q in the same way as OP above.Then ( ;U x m) (0 ) ~ 2-n.

' q

•

V.Je claim OPs U OPt

~

Oq ^c. OPt U OPs U

9

^(f)

where

(J

(f) denotes the graph of f, ( ,# x m) (

(j-

(f)) ^{=0 •} Proof: Let

<x,

r) ~ OPt

u

OPs , say

<

x, r ) ~:= OPt • Since

min(f,pt) a. min(f,q) and max(f,pt) £ max(f,q) ,

We have by definition that (x,r) ^E Oq.

Now let (x, r'> E Oq , and assume f(x)

1:

r. We may, by symmetry, assume f(x) ·

>

r. Then, for some a r= q, we have

r < a(x). But a will either be in Pt or in p~ , so

<

x,r')will be in 0 or 0 • But then

Pt Ps

( Jli

.'< m) (

o u op )

~ 2-n.

Pt s

We may then apply lemma 2 on the Opt's, and we get that there are only countably many of them.

If Pt

1:

Ps , t,s ⁶ T , then OPt

1:

OPs by the argument above. Thus, there are at most countably many Pt's , and the countable antichain condition is verified.

Now, by MA, let G be generic with respect to As in theorem 3 we may conclude that

J (

sup(f,G) - inf(f,G)) dp· ~ 2-n K

. l

-~IQ...,. _{- ~ .) ~ (:}

r·

For each n ^E IN, let Gn be as constructed above. Let ai n

E:-~

^{() Gn •}

Let

~ =

^{inf( am} CJ

,

^{0 <' ·' J,m / ' ~'. <i,n}

>

⁾

'

n sup( ^m

.

_<.J,m) ::: _<i,n) ) gi

=

aJ

,

Then

~ ,

^{-g. n} _~ ^6-

^s

_j:f!'].

~_ ~_)(,J..,n; ~I XIN is an increasing net

r n decreasing

-! gi \ /i n) c I>< IN is a net.

L. - <.... '

Let

cp'

^{= sup}

rtf . _,

_i ^<;:; _{I, nG IN}

₁

_•

Let

·'l.f ' =

^{inf . n ) g.} _{l. ~}

,

⁰ i G I, ^{n (;} lNJ _•

We call

if

^I = lim inf af!-

~

'1(• ^I ₌ lim sup af!-

~

Obviously ^{f'j; 9}

'r

^{:!f /l( ' •} We'll prove that

Cf

^!S

cp' , '-( '

^:1£ "'~.}/ will then follow by symmetry.

Let x ⁶ K, n ^& IN. We'll prove ^{rp'(x) ~} cf(x) - 1/n.

There will be an fi in the original net such that fi (x)

> cp

^{(x) - 1/2n}

Also for any ^{k ~ IN, j ~} i

\ m( ) • ... a. x ,

l. J

But then ffn(x) ~ Cf'(x) - ¹ /n • But

(f'

^{(x) ::- sup} { ffn(x) ;

<

i, 2n

'>

(p

^{'(x) ~}

cf'

(x) - 1 In •

E I X IN ~~

J

~ fi(x) - 1/2n

, so

To show

J ( ^{-·'r' - cp ')}

^d./.A

^{= o ,}

let n be given. It is K

sufficient to show that

/ \ max gn mJ.·n ....n d ,^{1 ' ..:::_} 22-n , or as well

J . -

Ij_ ^fVL

K i~I ~ ib-I

{

.. )

K

I

^{f -} min

ft.l I

i&-I ~ ^d/{...(.

But we have

j l f - min fn _i

I

^{d)A : L / )}

J

^{If - min a~} _~ ^{d!Vl ~}

.i

^{2-m = 21-n}

iEI i~I ^I

K m~n K ^{n1 '.;.I'J}

This ends the proof of lemma 6.

We can now give our main result:

Theorem 7:

Assume MA. then problem 1 has a positive solution.

Proof: Well-order all measures

,«

on K by a minimal wellordering,

lf"".L-><

^< ₂^t'Y'0 • Then each proper initial segment has cardinality..:::

r"'<Q

2 •

By induction on the ordinal ·>\ define two sequences ^{·i· /rJ l} and

l '-r '-'( j

such that i)

ii) At limit stages

iii) At sucessor stages ,) ⁽

K

I cp ~

⁺¹

iv) For all ,--< :

(f , - '1-'o( ( 2.

""'

Then, by theorem 3, each

(p _,

_,:--, and ~,.:: are universally measurable by MA.

Use lemma 5 to obtain ii) , lemma 6 to obtain iii).

Let F

=

sup _~-

=

inf ^{. ··1-f' '}

?(' ^->{ ,:v'o

"'

·-< < 2 0 _{~-" <} 2

=

⁰

To see that F is determined at a point x, let

;t--1 .

be the point

J

r

measure of x. Then

cp

^{(>+1 (x)}

= t

~· ^{+1 (x)} by ii).

To show F is universally measurable, let /'{ (j be any

measure. Then F

= q~ ~-~

⁺¹

=

.,_r· (·. +1 almost everywhere and both

(e

_{... :;+1} and

y

_t'+1 ^are _{IM c.} -measurable.

F is affine since F is both concave ( sup

con vex ( inf "'({ ) • Since lim

i

^{0 ~' n E IN}

=

0, we have F( -f 0 \ ,:;_ IN) = 0. Thus F is linear.

·- vn-... ^I

) and

(/)('> ),

In Hokobodzki 's original proof he defines L ( here denoted

2 r.

1 ) to be limits of increasing sequences from S. Then he can replace all nets by sequences, and in lemma 6 he can repl2ce our

Hf

^{by JUSt ~} , i.e. a decreasing sequence. Then the point where we used MA will be trivial~) JUSt find one .function a 6 Hn (=~ ) •

This is the only point where we have added anything new to the proof.

Vlill 2. M

= )

^? The answer is ¹¹no¹¹ if CH fails.Note that all elements of 2 M are Borel. Let K1

= Co,

¹ l IN. Let Q be an

;\A1-set on the extreme boundary of

Co,

¹

J

IN that is linearly independent. Define

( 1 if x is generated from ^Q F(x) = .!

l_

0 otherwise

Then F {_:..

2

on K1 • If F is Borel, then

q

^(F)

⁼

the graph of F will be Borel and

Ci

(F) /) the extreme boundary of K1 will

ft

be Borel. But this set has cardinality /Y~_, which gives a

contradiction. ( All Borel sets are either countable or has the cardinality of the continuum.)

Theorem 8:

If problem 2 has positive solution by CH, then problem 2

l~~'S" positiv;.;: .solution in g.eneral.

The proof is JUSt by putting together available information.

We will state the general results and give references.

ZF

=

Zermelo-Fraenkel set theory.

Proposition 1 G~del (5)

Let V be the universe of sets. Assume V satisfies ZF. Then there is a subclass L .~ V such that

1. L satisfies ZF, CHand the axiom of choice.

2. L contains all ordinals.

Now let N

=

^JN)N with product topology. We may index the Borel-sets of K by elements from N, tBx ~. _{-' X E} A ,.- N • _':::::. vfe say that Bx is a Borel-set from L if the index x is in L.

The indexing has the following properties

i) A

=

the set of indices

=

the co-proJection of a Borel set from L. ( V!e have similar indexings for all Polish spaces.) ii) There is a set B '~ K xN, being the proJection of a Borel

set from L, such that whenever x is an index:

Proposition 2 Shoenfield (6)

Let B be Borel. Assume B .,;;:,_ K₁ ^l) K2 ' .... )K3 where the Ki 's are the spaces ^K or N. Define ^C by

X 1:= C <'-~ 1/ y 3 z ( ,:( x, y, z ) t: B ) Then, if x E- L, we have

x t= C <::-.> V y ^G L j z c=, L (

<

x, y, z

>

^{e B ) •}

Proof of theorem 8: Assume that we in L may prove that there is a Borel-function F with Borel graph

0

(F). This will have index i in L.

1. Bi is a graph in L <~> Bi is a graph.

Proof·: V ^X 3_ y ( < x, y ) ^·f- B.). Here use proposition 2.

~

\fx Vy l(z (

<

x,y; (; B;& <x, z) rc B-

I y

=

z ) •

For simplicity, use proposition 2, though we here are in a simpler case.

2. Bi is linear in L <.--; Bi is linear.

Proof: Bi is linear

( 10"' X \/ y k z ii u l7 v \7'' 1!! ) (

<

^{x, u) 6: B,} & ,:_ y' v '> ⁶ Bi &

( Z, 1.V ') G B~ & z = X $ y -=-~ W = u + v ) operation on K.

Analogous with mult~plication.

where (:£) is the

Also here proposition 2 may be used.

3. Bi is medial in ^L <::_.:.--:..-;; Bi is medial.

Proof: t/ ^{< (} x. ', . ^-, _IN ^+;ly

((

c. ~ '~ f

.. t~ lim sup xi ⁾

Here the xi are taken from JR.

Also here use proposition 2.

:; x. -~; . . IN' y) (, B .::) lim inf x. ~

. ~. ~ {": ~

That the quantifiers are taken over diffurent dimensions does not matter as long as all spaces are Polish spaces. ( See Normann (7))

Bibliography:

P..na.lysis:

(1) Mokobodzki, Qabriel: Fonction medial d'une suite de fonctions. £1imeograph, 6 pages. Paris.

(2) Alfsen, Erik M.: Compact convex sets and boundary integrals.

Springer Verlag. 1970.

Besides (1) see also

(1a) Mokobodzki, Gabriel~ Ultrafiltres rapides sur 1N. Construc~ion

d'une densite relative de deux potentiels comparables.

( Seminaire Brelot-Choquet-Deny- 12e annee 67/68 n° 12.

Institut Henri Poincare Paris.)

(1b) Mokobodzki, Gabriel: Quelques propriet~s des fonctions numeriques convexes ( s.c.i. ou s.c.s.) sur un ensemble convexe compact. ( Seminaire Brelot-Choquet-Deny. 6e annee 1962. n°.9. Institut Henri Poincare Paris.)

Logics.

(3) Solovay-Tennenbaum: Iterated Cohen extensions and Souslin's problem. J\nnals of Mathematics. 94(1971) 201-245.

(1+) ~l!artin-Solovay: Internal Cohen extensions. ^Ann. of Math.

Logic. 2 (1970) 143-178.

(5) Gddel: The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory. Ann. of Math. Study No.3. 1940.

(6) Shoenfield: The problem of predicability. Essays on the fund. of math. Jerusalem 1961, 132-39.

(7) Normann, Dag: :r.1alteori og logikk. (Hovedfagsoppgave 1972.)