La seguridad juridica

Note that each of the objectsP_ma, O_(ν),Xa

m defined in Section3.2depends on a game

G∈Θ_{. In this section, we will letP}a

m(G),O(G,ν),Xam(G) denote these objects to indicate

the dependence onG.

By the Existence Theorem3.2and Theorem3.4, for each gameG∈Θ_{there is a com-}

plete type structure T=

D

Sa,Sb,Ta,Tb,λa_G,λb_GE

such thatR_∞a(G,T_{) andR}b_∞(G,T_{) are nonempty. Our task will be to choose such maps} λ_Ga,λb_G for eachG∈Θ_{so that (G,u}a₎7→λ_Ga(ua) is a Borel map fromΘ×Ta intoL_(Sb×

Tb), and similarly witha andb reversed. If the set of gamesΘ_{were countable, then we}

could directly appeal to the Borel Isomorphism Theorem and glue the mapsλGtogether.

However, we will need to choose the mapsλG more carefully sinceΘis uncountable.

The following lemma improves Theorem3.2by specifying in advance the length of λa(ta) for each typeta∈Ta. For the remainder of this section, letM= |Sa| + |Sb|.

Lemma G.1. Let Ta, Tbbe uncountable Polish spaces and let©T_na:n>0ªand©T_nb:n>0ª

be countable partitions of Ta, Tb. For each game G∈Θ_{, there exists a complete one-to-one} lexicographic type structureT=­Sa,Sb,Ta,Tb,λa,λb®such that for each k≥M+1,

R_∞a(G,T₎∩(Sa×T_ka)6=∅, R_∞b (G,T₎∩(Sb×T_kb)6=∅;

and for each k>0, ta∈T_ka, tb∈T_kb,λa(ta)andλb(tb)have length k.

Proof of LemmaG.1. The proof is a routine modification of the proofs of Lemma3.15

and Theorem3.13so that for eachk>0, types inT_kaare mapped to LPS’s of lengthk. By the method of Lemma3.15, one can build a family of sets

{Q_ma :m>0}, {Q_mb :m>0}

such that(i)–(vi)of Theorem3.13hold withinT_ka for eachk≥min(m,M). That is, we have the following for each nonemptyXa⊆Saand eachk>0, and similarly forb.

(i) ©Γa_(Xa_,Qma₎∩T_ka:m>0ªis a decreasing chain of Borel subsets ofT_ka; (ii) For eachm>0,Γa_(Xa_,Qma₎∩T_ka6=_∅ ⇐⇒ ¡Xa∈Xa

m∧k≥min(m,M) ¢ ; (iii) Γa_(Xa_,Qa_∞)∩T_kais dense inΓa_(Xa_,Qa M+1)∩T a k, (iv) Γa_(∅,Qa 1)∩Tkais uncountable; (v) IfXa∈Xa

mandk≥min(m,M) then

(Γa_(Xa_,Qma_{) \}Γa_(Xa_,Qa

m+1))∩T

a k

is uncountable, and ifm<M thenΓa_(Xa_,Qa

m+1) is not even dense inΓ

a_(Xa_,Qa m);

(vi) IfXa∈Xa

∞andn≥M thenΓa(Xa,Q∞a )∩Xnais uncountable;

Condition(v)is upgraded to insure that fork≤M, no LPS inL_(Sb×Tb) of lengthkcan assume all ofQ₀a, . . . ,Qa_k. Then each piece ofT_ka will have the same cardinality of the corresponding piece ofL_k(Sb×Tb). The Borel Isomorphism Theorem can now be used as in the proof of Theorem3.13to construct the required mappingsλaandλb.

Next, we show that the gamesG∈Θ_{can be classified into finitely many shapes. We}

say that two gamesG,H∈Θ_{have thesame shapeif}Xa

for allm. By Theorem3.8, ifG andH have the same shape, thenS_ma(G)=Sa_m(H) and

Sb_m(G)=Sb_m(H) for eachm.

The next lemma shows that the sequencesSa_m(G) andXa

m(G) stabilize atM = |Sa| +

|Sb|, and hence there are only finitely many possible shapes of games inΘ_.

Lemma G.2. For each G∈Θ_{and m}≥M we have

(i) Sa_m(G)=Sa_M(G)=S_∞a (G)and Sb_m(G)=Sb_M(G)=Sb_∞(G); (ii) Xa

m(G)=XaM+1(G)=X∞a (G)andXbm(G)=XbM+1(G)=Xb∞(G).

Hence there are only finitely many shapes of games inΘ_.

Proof of LemmaG.2. Proof of (i).IfS_ma(G)=Sa_m+1(G) andS_mb(G)=Sb_m+1(G), then we see from the definition ofS_ma(G) thatS_ma(G)=Sa_n(G) andSb_m(G)=Sb_n(G) for alln≥m. More- over,S₀a(G)=SaandS₀b(G)=Sb, and the setsSa_m(G),S_mb(G) decrease withm. Therefore, the pair of sets (Sa_m(G),Sb_m(G)) can change at mostMtimes, and(i)follows.

Proof of (ii).Letm>M andXa∈Xa

m(G). ThenXa=O(G,µ) for someµ∈Pma(G). We

haveµ=νν′for someν∈N _(Sb_{) with Supp(ν)=S}b

m−1(G) and someν ′_∈Pa

m−1(G). By 1.,

Sb_m(G)=Sb_m−1(G), soµ′=νµ∈P_ma(G). It is clear thatO_(G,µ′₎=O_{(G,µ), soX}a∈Xa

m+1(G). This proves(ii).

LemmaG.2shows that the shape ofGdepends only onXa

m(G),Xbm(G) form≤M+1.

We may therefore define theshape ofGas follows. Given a sequence

S=(Xa

1, . . . ,XaM+1,X

b

1, . . . ,XbM+1),

we say thatG has shapeS_{, and write} S_(G)=S_{, if}Xa

m =Xma(G) and Xbm =Xbm(G) for

m=1, . . . ,M+1. And we say thatS_{is agame shapeif there exists a gameG}∈Θ_{such that}

S=S_(G).

The intuitive idea of our proof of Theorem F.1 will be to build the type structures T_{G in such a way that they can be glued together by an inductive construction on the} length of LPS’s. For each fixed lengthk >0, we will see that the setΘ _{of games can be}

partitioned into finitely many classes such that within each class, the lengthk parts of the type structuresT_{G can be chosen to be the same up to a Borel transformation, and} thus can be combined into a single type structure.

To do this, we will need some results from the literature about definable sets in the ordered field of real numbers. We letF= 〈R_{, 0, 1,+,·,}<〉be theordered fieldof real num- bers. A set ofn-tuples A⊆Rn _{is said to bedefinable(in}F_{) if Ais the set of alln-tuples} that satisfy a first order formulaϕ(x1, . . . ,xn,~c) inFthat has the variablesx0, . . . ,xnand a

finite tuple~_{cof parameters in}R_{. Given two definable setsA}⊆Rm_,B⊆Rn_inF_{, a function}

f :B→Ais said to bedefinable(inF_{) if its graph}©₍~_x,~_{y) :f(}~_x)=~_yª_{is definable.}

The celebrated classical result ofTarski(1951) shows that a set is definable inF_{if and} only if it is semi-algebraic (i.e., definable by finite collections of equations and inequali- ties between polynomials). Tarski’s theorem has the following easy consequence. Proposition G.3.

(i) F_{is o-minimal, that is, every set A⊆}R_{that is definable in}F_{is the union of finitely}

many open intervals and singletons.

(ii) Every set A⊆Rk_{that is definable in}F_{is Borel.}

We refer to the monograph van den Dries (1998) for an exposition of o-minimal structures, but we will only need the particularo-minimal structureF_{. We will need a} result ofHardt(1980), which says that every definable function can be partitioned into finitely many definable pieces that each look like the projection of a product of two sets onto one of the factors. This result was generalized too-minimal structures (seevan den Dries,1998, chap. 9, Theorem 1.2).

Definition G.4. Suppose A⊆Rm_,B ⊆Rn_{, g :B} → A is definable, and g maps B onto A. We say that g isdefinably trivialif there exists a definable set C⊆Rk_{for some k and a de-}

finable function h:B→C such that the function(g,h) :B→A×C is a homeomorphism.

Proposition G.5(Hardt(1980)). Let A⊆Rm_,B⊆Rn_{, and g:B}→A be definable, and sup- pose g maps B onto A. Then there exists a finite partition©A1, . . . ,Ap

ª

of A into definable sets such that for each i≤p, the restriction of g to g−1(Ai)is definably trivial.

In the above proposition, note that for eachi, the setBi=g−1(Ai) and the restriction

gi of g to Bi are also definable. The result says that there is a definable set Ci and a

definable functionhi :Bi→Ci such that (gi,hi) :Bi→Ai×Ciis a homeomorphism, i.e.,

one-to-one, onto, and bi-continuous.

We now look at definable properties of games and tuples of probability measures. Recall thatM_(Sb_{) is the set of all probability measures onS}b_{. We may identify a prob-}

ability measureν∈M_(Sb_{) with the real vector}〈ν(sb) :sb∈Sb〉 ∈R|Sb|_{and note that this} tuple satisfies the first order formulas 0≤ν(sb)≤1 andP_sb_∈Sbν(sb)=1. Similarly, for

each fixedk, thek-tuple of probability measuresν∈N_k(Sb_{) is identified with ak}· |Sb|- tuple of reals in the obvious way.

By a k-fold support inSb we mean a k-tuple Y =(Y0, . . . ,Yk−1) of nonempty sets

Yi⊆Sbsuch thatSj<kYj=Sb. Thek-fold supportof ak-tuple

ν=(ν0, . . . ,νk−1)∈Nk(Sb)

is thek-tuple Supp_k(ν)=¡Supp(ν0), . . . , Supp(νk−1)

¢

.

The next lemma shows that for each fixedk, certain relations involving games and

k-tuples of probability measures onSbare definable.

Lemma G.6. For each k, the following sets are definable. (i) For each n, the set©(G,ν)∈Θ×N_k(Sb_{) :ν}∈P_na(G)ª;

(ii) For each k-fold support Y in Sb, the set©ν∈N_k(Sb_{) : Supp}

k(ν)=Y

ª ; (iii) For each Xa⊆Sa, the set©(G,ν)∈Θ×N_k(Sb_{) :}O_(G,ν)=Xaª; and

(iv) For each game shapeS=¡Xa

1, . . . ,XaM+1,Xb1, . . . ,XbM+1

¢

, the set{G∈Θ_:S_(G)=S_}.

Proof of LemmaG.6. Proof of (i–iii). These can be seen by writing the definitions for- mally in first order logic.

Proof of (iv).By LemmaB.4, for eachnand each setXa⊆Sa, we haveXa∈Xa

n(G) if

and only if there exists anν∈N_M+1(Sb₎∩P_na(G) such thatO_(G,ν)=Xa. The point is that we need only considerν’s of lengthM+1. The result now follows from(i)and(iii).

Definition G.7. A k-good partitionofΘ_{is a finite partition}©_{A1, . . . ,Ap}ª_ofΘ_{such that} for each i≤p,

(i) Ai is definable;

(ii) ∃S_i∀G∈Ai, S(G)=Si, i.e., all the games in Ai have the same shape;

(iii) For each set X ⊆Sa and each k-fold support Y in Sb, the projection function from the set

Bi,X,Y ={(G,ν)∈Ai×Nk(Sb) :O(G,ν)=X∧Y =Suppk(ν)}

to Ai is definably trivial.

Remark G.8. For each k-good partition ofΘ_{, the family of sets}

{Bi,X,Y :i ≤p∧X ⊆Sa∧Y =Suppk(ν)}

indexed by(i,X,Y)in(iii)is a finite partition ofΘ×N_k(Sb_{)into definable sets. The set}

Bi,X,Y may be empty for some values of(i,X,Y).

Lemma G.9. Suppose©A1, . . . ,Ap

ª

is a k-good partition ofΘ_{. Let i}≤p; Gi∈Ai, X⊆Sa; Y

©

ν∈N_k(Sb_{) : (Gi,ν)}∈Bi,X,Yª.24 Then there is a definable function h:Bi,X,Y →Ci,X,Y such

that the function(g,h) :Bi,X,Y →Ai×Ci,X,Y is a homeomorphism.

Proof of LemmaG.9. By(iii)in DefinitionG.7, the projection functiong fromBi,X,Y to

Ai is definably trivial, so there is a setD and a definable function f :Bi,X,Y →D such

that the function (g,f) :Bi,X,Y → Ai×Dis a homeomorphism. Then the restriction of

f to {Gi}×Ci,X,Y is a homeomorphism from {Gi}×Ci,X,Y to D. Therefore, there is a

definable homeomorphismℓfromDtoCi,X,Y, and hence the compositionh=ℓ◦f has

the required properties.

PropositionG.5gives us the following lemma about game-LPS pairs. Lemma G.10. For each k>0, there exists a k-good partition ofΘ_.

Proof of LemmaG.10. By LemmaG.6, for each game shapeS_andk>0, the sets

AS={G∈Θ:S(G)=S}, BS=AS×N_k(Sb)

are definable. It suffices to prove that for each game shapeS_{, the set A}Sadmits a finite

partition into definable sets©A1, . . . ,Ap

ª

such that(iii)holds for eachi ≤p. If so then the union of these partitions will be ak-good partition ofΘ_.

Now fix a game shapeS_{, and letgbe the projection function fromBSontoAS. Since} the setsSa andSb are finite, there are only finitely many (X,Y) such thatX ⊆Sa andY

is ak-fold support inSb. For such (X,Y), let

BX,Y =

©

(G,ν)∈BS:O(G,ν)=X and Supp_k(ν)=Y

ª

.

By LemmaG.6, each setBX,Y is definable, and hence the restriction ofgtoBX,Y is defin-

able. By PropositionG.5, there is a finite partition©A1,X,Y, . . . ,Aq,X,Y

ª

ofASinto defin-

able sets such that for each j≤q, the restriction ofg to (g−1(Aj,X,Y))∩BX,Y is definably

trivial. Let us say that two gamesG,H∈Aareequivalentif

© (j,X,Y) :G∈Aj,X,Y ª =©(j,X,Y) :H∈Aj,X,Y ª .

There are finitely many equivalence classes in AS and each equivalence class is defin-

able. Therefore, this equivalence relation implicitly defines a finite partition of ASinto

definable sets©A1, . . . ,Ap

ª

and this partition satisfies(iii)as required. We are now ready to prove TheoremF.1.

24_i.e.,C

Proof of TheoremF.1. We will construct a pair of Borel functions

κa:Θ×Ta→L_(Sb×Tb), κb:Θ×Tb→L_(Sa×Ta)

with the required properties in several steps. Steps 3 and 5 will require additional proof. 1. First, choose partitions©T_ka:k>0ªand©T_kb:k>0ªofTaandTbinto continuum-

large Borel sets so thatTa=U_k>0T_kaandTb=U_k>0T_kb.

2. By LemmaG.10, we can choose ak-good partition©A1,k, . . . ,Ap(k),k

ª

ofΘ_{for each} k. Recall that for alli, games inAi,khave the same shape.

3. Next, for each (i,k), we construct a Borel mapκa :Ai,k×T_ka→Lk(Sb×Tb) such

that for allG∈Ai,k,κa(G,T_ka)=Lk(Sb×Tb). We will subdivide the domain even

further in this step.

4. By joining such maps for alli≤p(k), we will get a Borel mapκa:Θ×T_ka→L_k(Sb_×

Tb) sincep(k) is finite. Finally, we will join such maps for allk∈N_{, to get a Borel} mapκa:Θ×Ta→L_(Sb×Tb).

5. Lastly, we will verify thatκaandκbsatisfy the desired properties. Step 3.We begin by fixingk, ak-good partition©A1,k, . . . ,Ap(k),k

ª

ofΘ_{, andi}≤p(k). By Lemma G.1, for each gameG ∈Θ_{, we can choose a complete one-to-one type}

structureU_G=­Sa,Sb,Ta,Tb,λa_G,λb_G®such that for eachj≥M+1,

R_∞a(G,U_G)∩(Sa×T_ja)6=∅, R_∞b (G,U_G)∩(Sb×T_jb)6=∅;

and for eachj>0,ta∈T_ja,tb∈Tb_j,λ_Ga(ta) andλb_G(tb) have length j.

If we could glue together the mapλ_Ga for eachG∈Ai,kto define a Borel map (G,ta)7→

λG(ta) then we would be done. However, there are uncountably manyG’s inAi,k, so we

cannot appeal to the Borel Isomorphism Theorem.

In order to get around this problem, we fix someGi,k∈Ai,k and the associated type

structureU_G

i,k. For the sake of avoiding subscripts of subscripts, we will letUi,k=UGi,k,

λa_i,k=λ_Ga

i,k, andλ b i,k=λ

b

Gi,k. We will soon show that the structural properties shared by

the games in Ai,k allow us to define a Borel mapκaonAi,k×T_kafrom the mappingλa_i,k

so thatκahas the desired properties.

For eachX ⊆Saand eachk-fold supportY inSb, let

Bi,X,Y,k= n (G,ν)∈Ai,k×Nk(Sb) :O(G,ν)=X∧Suppk(ν)=Y o ; and Ci,X,Y,k= n ν∈N_k(Sb_{) : (Gi,k,ν)}∈Bi,X,Y,k o as in LemmaG.9; and Di,X,Y,k= n ta∈T_ka: marg_Sb(λ_ia_,k(ta))∈C_i,X,Y,k o .

Note that the setsBi,X,Y,kandCi,X,Y,kare definable. By PropositionG.3,Bi,X,Y,kis Borel,

and henceDi,X,Y,k⊆T_kais Borel as well. We note that for eachi≤p(k), the family of sets

n

Di,X,Y,k:X ⊆SaandY is ak-fold support inSb

o

is a partition ofT_kainto finitely many Borel sets, some of which may be empty.

We will define the restriction ofκatoAi,k×Di,X,Y,k. We fixX ⊆Sa, and ak-fold sup-

portY inSb. Letg be the projection function fromBi,X,Y,ktoAi,k. By LemmaG.9, there

is a definable function h :Bi,X,Y,k →Ci,X,Y,k such that the function (g,h) :Bi,X,Y,k →

Ai,k×Ci,X,Y,kis a homeomorphism.

Since Y is a k-fold support in Sb, we can write Y =(Y0, . . . ,Yk−1). Now, let LY =

© µ∈M_(Sb×Tb) : Supp_kmarg_Sbµ=Y ª and MY = © ν∈M_(Sb_{) : Supp} kν=Y ª . Let φY :

LY ×MY →LY be the function that maps (µ,ν) toφY(µ,ν) such that the j-th compo-

nent of [φY]j is defined as follows for eachj <k.

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