On $\Game\Gamma$-complete sets

ON a -COMPLETE SETS

GABRIEL DEBS AND JEAN SAINT RAYMOND

Abstract. Extending a result of A. Kechris we prove that under suitable assump- tions on the class of Borel sets, in particular when is a Baire class, if anya set is reducible to somea setAby aa -measurable function thenAisa -complete.

In all the sequel by aclass⇤we mean a collection of subsets of Polish spaces closed under taking inverse images by any continuous function f : X ! Y between Polish spaces. We denote by ˇ⇤ the dual class of⇤, that is the class of all complements of sets in⇤.

We recall that, given a class⇤, a subset A of some Polish space X is said to be⇤- hard if any set B ⇢2^! in⇤is reducible to A by a continuous function, that is if there exists a continuous functionf : 2^!!X such thatB =f ¹(A). If moreoverA is in ⇤ then A is said to be ⇤-complete. In [8] Kechris introduced the more general notion of F-⇤-hard(complete)set by requiring the reduction functionf to be in some given class of functionsF. WhenF is the class of all Borel functions he proved the following:

Theorem 1. (Kechris)If a setA is Borel-⇧¹₁-hard thenA is⇧¹₁-hard.

From this follows that if a setA is Borel-⇧¹₁-complete then Ais ⇧¹₁-complete. Note that, as pointed out by Kechris in [8], the proof of this classical result makes use of methods from E↵ective Descriptive Set Theory, namely through the use of a coding lemma from [5] which relies on the recursion theorem.

More recently (see [2] and [3]) evaluating the complexity of some specific sets beyond the class ⇧¹₁ we were led to situations where the natural reduction functions are non Borel, though in each case we were able, at the price of some additional work, to con- struct continuous reductions. In fact, as pointed out by Kechris in [8], under suitable determinacy assumptions Theorem 1 extends to higher projective levels. For example under Det( ¹₂) if F is the class of all ¹₂ functions then any set which isF-⇧¹₂-hard is

⇧¹₂-hard; and it is easy to see that the determinacy assumption in this statement cannot be dropped. However our goal in the present work is to prove in ZFC a version of Theo- rem 1 which applies to classes⇤ beyond⇧¹₁, still strictly below ¹₂, replacing the class of Borel functions by the class of all⇤-measurable functions.

We recall that a partial functionf :D⇢X!Y with an arbitrary domainD(subset of some Polish spaceX) is said to be⇤-measurableif for any open subsetV ofY there exists a subsetA ofX in⇤ such thatf ¹(V) =D\A. Since any class is closed under discrete unions (i.e. unions of the form S

i2IAi with for all i,Ai 2⇤ andAi ⇢Ui for some family(Ui)_i2I of pairwise disjoint open sets)the notion of⇤-measurable function coincides with the notion of⇤-recursive function in the sense of [10].

Note that the composition of two partial⇤-measurable functions is not⇤-measurable in general, unless the class⇤has thesubstitution property(see [10]); and we shall say that

2010Mathematics Subject Classification. Primary 03E15, 28A05; Secondary 54H05.

Key words and phrases. ⇤-hard,⇤-complete, Game operator.

1

⇤ has the weak substitution property if the composition f g of any total⇤-measurable function f : X ! Y with any partial ⇤-measurable function g : D ⇢ Y ! Z (with X, Y, Z Polish spaces) is ⇤-measurable, equivalently if the inverse image by any total

⇤-measurable functionf :X !Y of any⇤ subset ofY is in⇤. In fact even the weak substitution property is a very strong closure property satisfied by very few classes, and the only known examples of such classes are of the form ⇤ = a where is a (non arbitrary) class of Borel sets andais Moschovakis game operator (see Section 1 for more details).

To state simply our main result we introduce the following notion:

Definition 2. A setA is said to beweakly⇤-hardif any setB ⇢2^! in⇤ is reducible toA by a⇤-measurable functionf : 2^!!X.

Thus the question we are addressing is to find sufficient conditions on a given class⇤ so that any weakly⇤-hard set is⇤-hard. Observe that at least in the non self-dual case ( ˇ⇤6=⇤), and assuming enough determinacy, such a class⇤ (equivalently the dual class

⇤) should necessarily have the weak substitution property. Indeed, suppose that thereˇ exists some setA2⇤ˇ and a total⇤-measurable functionf such thatB=f ¹(A)62⇤,ˇ then by Wadge determinacy up to the class of the set B, any set in ⇤ is reducible to B by a continuous function. So, by composition, any set in ⇤ is reducible to A by a

⇤-measurable function. HenceA is weakly⇤-hard but since it is in ˇ⇤the setA cannot be⇤-hard.

This latter observation leads us naturally to restrict our investigation toa –classes. In fact the statement of the main theorem requires some additional assumptions on the class which we will introduce in Section 1. We only mention here the following particular case.

Theorem 3. If is any Baire class (i.e. ⌃⁰_⇠ or⇧⁰_⇠ for some ⇠2!₁) then any weakly a -hard set isa -hard.

In particular since ⇧¹₁ =a⌃⁰₁ and total ⇧¹₁-measurable functions are just the Borel functions, then Theorem 1 is a particular case of Theorem 3.

To finish we give next an application of Theorem 3 connected to the concrete situations encountered in [2] and [3], and which were our initial motivation for this work. Let us first recall that ifX is a Polish space there is no canonical Polish topology on the space F(X) of all closed subsets ofX. HoweverF(X) can be endowed with a canonical Borel structure, called theE↵ros Borel structure, which is standard, that is Borel isomorphic to the Borel structure on 2^!(see [7], Section 12C). This is the Borel structure generated by the sets of the formF⁺(V) ={F2F(X) : F\V 6=;}whereV is an arbitrary open subset of X. Hence if is a class closed under Borel isomorphism one can can speak about -hard(complete) subsets ofF(X) to mean -subsets which are -hard(complete) relatively to some (equivalently any) Polish topology onF(X) defining the E↵ros Borel structure.

Corollary 4. Let X and Y be Polish spaces, E a ⌃¹₁ subset of X⇥Y andA⇢F(Y).

For all⇠ 2if the set {x2X : E(x)2A} isa⌃⁰_⇠-hard then Aisa⌃⁰_⇠-hard.

Proof. Let :X !F(Y) be the function defined by (x) =E(x). Observe that since E is⌃¹₁the set F ={(x, y)2X⇥Y : y2E(x)}is⌃¹₁ too. Hence for any open subset V of Y the set ¹(F⁺(V)) is⌃¹₁ as the projection on X of the set F \(X⇥V). It follows that the function is measurable with respect to the smallest class containing

all⌃¹₁ sets and closed under countable unions and complementation, which is a subclass ofa ⁰2(see Proposition 1.1). In particular, since⇠ 2, isa⌃⁰_⇠-measurable, and since by assumption the set ¹(A) isa⌃⁰_⇠-hard, then given anya⌃⁰_⇠ set B⇢2^! we can find by Theorem 3 a continuous function f : 2^! ! X which reduces B to ¹(A), hence f : 2^!!F(Y) is aa⌃⁰_⇠-measurable function which reducesB toA. This proves that

Ais weaklya⌃⁰_⇠-hard henceAisa⌃⁰_⇠-hard. ⇤

1. Main Theorem

We introduce in this section the technical notions needed for the statement of the main result. We first recall some basic definitions and fix some notation.

Games: By a free game on a set⌦we mean a game in which at each move the players can choose any element in⌦. But we shall consider mainly games with givenrulesthat is games in which the finite runs are constraint to stay in a given treeT on⌦to which we shall refer as the frameof the game. The set of infinite runs in such a game is then the setdTeof all infinite branches ofT, and a win condition is a subset ofdTe. Note that several notions, for example the general notion of strategy, depend only on the frame of the game and not on any win condition.

We view a strategy, say for Player I, in a game with frameTas a function :⌦^<!!⌦ which to anyu2⌦ⁿ (viewed as the sequence of Player II’s moves in some virtual run of length 2n) assigns an element of ⌦(viewed as Player I’s next move in this virtual run).

More precisely for any 2!^! let ^{(0) (1)} denote the subsequence of of even (odd) coordinates, so: ⁽⁰⁾(n) = (2n) and ⁽¹⁾(n) = (2n+ 1). A run v 2 T is said to be compatible with if for all m < |v|

2, v(2m) = (v_|2m⁽¹⁾), and is a strategy if for any run of even lengthv2T compatible with ,w=v^_h (v⁽¹⁾)iis a run in the game, (i.e.

w2T).

Two gamesGandG⁰ are said to be equivalent (and we writeG⇡G⁰) if whenever one of the players has a winning strategy in one of the games, the same player has a winning strategy in the other game.

We shall also consider specifc games for which the frame tree is the treeTRof all finite R-chains for some giventransitive relationRon⌦; and we shall then say that the game is atransitive game. Note that by transitivity, in such a game the validity of a move for any of the players depends only on his opponent’s last move, and any subsequence of a given run can also also be viewed as a run in the game.

The game operator: For any setA⇢⌦^!we denote byG_Athe free game on⌦in which the win condition for Player I is given by the setA. We recall that ifB⇢X⇥!^! then

aB:={x2X : Player I has a winning strategy in the gameG_B(x)}

and for any class , a denotes the class of all sets of the formaB with B in . For any class ,a is closed under countable unions and intersections and if ⇢ ¹1 then by determinacyaˇ is the dual class ofa .

The reader will find in [10] a detailed account ona classes. Let us only mention that a⌃⁰₁=⇧¹₁,a⇧⁰₁=⌃¹₁ and the following explicit description ofa( ⁰₂).

Proposition 1.1. a( ⁰₂)is the smallest non trivial class closed under complementation and the Suslin operationA(known as Selivanovski’s class ofC-sets).

More fundamental are the following two properties (see [10], Theorem 6D.3 and The- orem 6E.1), which will play a central role in our study.

Theorem 1.2. (Moschovakis) For all⇠ 1:

a) The classa⌃⁰_⇠ is normed and has the substitution property.

b) For any setB⇢X⇥!^!in⌃⁰_⇠ there exists aa⌃⁰_⇠-measurable function which assigns to any x2aB a winning strategy for Player I in the gameG_B(x).

Definition 1.3. We shall say that a class is a Moschovakis class if for any set B ⇢ 2^!⇥!^! in there exists a a -measurable function onaB which assigns to anyx2aB a winning strategy for Player I in the gameG_B(x).

We can now state our main result in its full generality. We recall that a class⇤is said to have the!-reduction propertyif for any sequence (An)_n2!in⇤there exists a sequence (Bn)_n2!of pairwise disjoint sets in⇤such thatBn ⇢AnandS

n2!Bn=S

n2!An. Note that if the class ⇤ is normed and closed under countable intersections then⇤ has the

!-reduction property. Indeed if':A!Onis a ⇤-norm on the setA=S

n2!A_n⇥{n} then one can take:

B_n={x2A_n: 8k < n,(x62A_k or '(x, n)<'(x, k)) and 8k > n,(x62A_k or'(x, n)'(x, k)}. In particular any classa which is normed has the!-reduction property.

Theorem 1.4. Let be a Moschovakis class of Borel sets such that the classa has the

!-reduction property and the weak substitution property. Then any weaklya -hard set is a -hard.

Hence Theorem 3 follows readily from Theorem 1.2 and Theorem 1.4. In a forthcoming note ([4]) we provide more classes of Borel sets satisfying the assumptions, hence the conclusion, of Theorem 1.2. In particular this is the case for all the di↵erence classes D_⌘(⌃⁰_⇠) if⇠ 2.

The rest of the paper is devoted to the proof of Theorem 1.4. The general scheme of this proof is strongly inspired by Kechris’ original proof of Theorem 1. However we will not use the coding lemma of [5]. Note also that if = ⌃⁰₁ the powerful Moschovakis general results on a are not needed and in this particular case Theorem 1.2 can be proved by elementary methods, which provides a classical proof for Theorem 1.

2. Non meagera sets

Notation 2.1. From now on we fix a Polish space X equipped with a complete metric d1 and (U_n)_n2!an enumerated basis of nonempty open subsets.

The Banach-Mazur game: We denote by/the transitive relation on! defined by m/n () U_m⇢U_n and diam(U_m) 1

2diam(U_n)

byT the associated tree of all finite/-chains and byG^⇤⇤ the transitive game with game treeT:

I : m₀ m₂ · · ·

G^⇤⇤: with m_n+1/m_n

II : m₁ m₃ · · ·

So for each infinite run↵= (mn)2 dTethe setT

nUmnis a singleton{x↵}; and since all the setsUmare open and diam(Umn)&0 then the function :dTe !X defined by (↵) =x↵is continuous.

For any setA⇢X we denote byG^⇤⇤_A the game in which the win condition for Player I is given by “x_↵2A”. It is well known that:

Proposition 2.2. For any setA⇢X:

a) Player I wins the gameG^⇤⇤_A if and only if A is comeager in some nonempty open subset ofX.

b) Player II wins the gameG^⇤⇤_A if and only ifA is meager inX. Hence if the gameG^⇤⇤_A is determined then:

c) Player I wins the gameG^⇤⇤_A if and only ifA is non meager.

2.3. The game G˜^⇤⇤: In [6] Kechris studied some transitive games in connection with a given forcing. To any such game with win condition in a he associates an equivalent game with win condition in . We explicit next this association in the simple context of the Banach-Mazur games.

LetG˜^⇤⇤ denote the game frame on!²:

I : (m₀, k₀) (m₂, k₂) · · ·

G˜^⇤⇤: with m_n+1/m_n

II : (m₁, k₁) (m₃, k₃) · · ·

while the second coordinatekn is free.

ThenG˜^⇤⇤ is also a transitive game and we denote by ˜T the corresponding rules tree on!², that we shall also identify to a subset ofT⇥!^<!where T is the Banach-Mazur game tree. We shall also identify any infinite run 2 dT˜eto a pair (↵, )2 dTe ⇥!^!. Then for any setB ⇢X⇥!^!we denote byG˜^⇤⇤_B the game with frameG˜^⇤⇤ in which the win condition for Player I is given by “( (↵), )2B” where : dTe !X is as in the Banach-Mazur game. Again any infinite subsequence ↵⁰ of a run ↵ is also a run with (↵⁰) = (↵), hence the winner of both runs is the same player.

Note that if B = A⇥{ 0} then, up to obvious identifications, G˜^⇤⇤_B = G^⇤⇤_A. More generally we have thegame formula([6]):

Theorem 2.4. (Kechris)Let A=aB.

a) If Player I wins the gameG˜^⇤⇤_B then A is comeager in some nonempty open subset of X.

b) If Player II wins the gameG˜^⇤⇤_B thenA is meager in X.

Hence combining Theorem 2.2 and Theorem 2.4 one gets:

Theorem 2.5. LetA=aB. If Player I (II) wins the gameG˜^⇤⇤_B then Player I (II) wins the gameG^⇤⇤_A.

In particular if the game G˜^⇤⇤_B is determined then the game G^⇤⇤_A is determined too and moreoverG^⇤⇤_A ⇡G˜^⇤⇤_B.

Next result provides, by a direct proof, a more precise version of Theorem 2.5 which we will need in the sequel. We state the result for Player I; the similar result for Player II also holds and is proved following the same arguments.

Theorem 2.6. There exists a continuous function : (!²)^((!2)<!)!!^(!<!) such that for any strategy for Player I inG˜^⇤⇤ :

a) ( )is a strategy for Player I inG^⇤⇤.

b) For any setB ⇢X⇥!^!, if is winning in the gameG˜^⇤⇤_B then ( )is winning in the gameG^⇤⇤_A whereA=aB.

Proof. For practical reasons we proceed to a change of notation by enumerating separately the moves of the players in the gamesG˜^⇤⇤ andG^⇤⇤ as follows

G^⇤⇤ : I : m⁰₀ m⁰₁ · · ·

II : m₀ m₁ · · ·

G˜^⇤⇤ : I : (m⁰₀, k⁰₀) (m⁰₁, k⁰₁) · · ·

II : (m₀, k₀) (m₁, k₁) · · ·

We recall that if 2⌦^!is a run in some game on⌦then ⁽¹⁾ denotes the sequence of Player II’s moves in this run.

Fix an enumeration (sn)_n2! of !^<! such that if sm ( sn (i.e. |sm| < |sn| and sm(k) =sn(k) fork <|sm|) thenm < n, sos0=;and|sn|n. For alln >0 set

`_n=s_n(|s_n| 1)

(i.e. `n is the last element ofsn). Letın :j7!nj be the canonical increasing embedding of|sn|inn+ 1 defined for allj <|sn|by

s_nj =s_n|j+1

soı_n(|s_n| 1) =nand for allj <|s_n|,s_n(j) =`_nj hence ifn >0 then:

sn= (`n0,`n1,· · · ,`n)

Given any : (!²)^<!!!² we shall define⌧= ( ) :!^<!!! by defining⌧(u)2! for allu2!^<!. Let 0 : (!²)^<! !! and 1 : (!²)^<!!! denote the first and second coordinate function of respectively. Then for allu2!ⁿ we define:

⌧(u) = ₀(u ı_n, s_n) (with the obvious identification (!²)ⁿ⇡(!ⁿ)²).

So⌧(;) = ₀(;,;)⇡ 0(;) and ifu= (m₀, m₁,· · ·, m_n)2!ⁿ⁺¹ then

⌧(u) = 0 (mn0,`n0),(mn,`n1),· · · ,(mn,`n) .

The function thus defined is clearly continuous, and we now prove parts a) and b) of the conclusion.

a) Suppose that is a strategy for Player I in G˜^⇤⇤ and let ⌧ = ( ). Observe first thatm⁰₀:=⌧(;) = 0(;) is well defined.

Consider then a nonempty run v = (m⁰₀, m0, m⁰₁,· · ·, m⁰_n, mn) in G^⇤⇤ compatible with ⌧, so satisfying for all p  n, m⁰_p = ⌧(m₀, m₁,· · ·, m_{p 1}), and let u = v⁽¹⁾ = (m0, m1,· · ·, mn) and m⁰_n+1 =⌧(u) = 0(u ın+1, sn+1). SinceG^⇤⇤ is a transitive game thenu ın+1is the sequence of Player II’s moves in a legal run inG^⇤⇤, hence (u ın+1, sn+1) is also the sequence of Player II’s moves in a legal run in G˜^⇤⇤ whose last element is (m_n,`_n+1), and since is a strategy inG˜^⇤⇤ then by definition of the rules in this game we have m_n/m⁰_n+1 where / is the transitive relation defining the game G^⇤⇤. Hence v^_hm⁰_n+1i 2T and this proves that⌧ is a strategy for Player I in G^⇤⇤.

b) Suppose now that is a winning strategy for Player I inG˜^⇤⇤_B, consider an infinite run

↵inG^⇤⇤ compatible with⌧ = ( ) and let x= (↵). We shall show thatx2A=aB, which is precisely the win condition for Player I inG^⇤⇤_A, by exhibiting a winning strategy

⌧xfor Player I in the free gameG_B(x)on!. A run compatible with⌧x goes as follows:

G: I : k⁰₀ k⁰₁ · · ·

II : k₀ k₁ · · ·

withk⁰₀= ₁(;) and for alln2!, if

(k₀, k₁,· · ·, k_n) =s_pn= `_(pn)

0,`_(pn)

1,· · · ,`_pn is nonempty then setting

u_n = (↵⁽¹⁾_|pn ı_pn, s_pn)

= (m_(pn)

0,`_(pn)

0),(m_(pn)

1,`_(pn)

1),· · ·,(m_pn,`_pn)

= (m_(pn)

0, k₀),(m_(pn)

1, k₁),· · ·,(m_pn, k_n) we define:

k⁰_n+1=⌧_x(k₀, k₁,· · · , k_n) := ₁(u_n).

Note that since↵is compatible with⌧ then

m⁰_pn+1=⌧ m₀, m₁· · · ·, m_pn = ₀(u_n) hence

(m⁰_pn+1, k⁰_n+1) = (u_n).

Consider now the following infinite run inG˜^⇤⇤_B

(m⁰₀, k⁰₀) (m⁰_p0+1, k₁⁰) · · · (m⁰_pn+1, k⁰_n+1) (m_p0, k₀) · · · (m_pn, k_n)

which is of the form (↵^⇤, ) with ↵^⇤ a subsequence of ↵, and an infinite run on ! compatible with⌧_x. Then since this run is compatible with which is a winning strategy for Player I inG˜^⇤⇤_B then ( (↵^⇤), )2B, and since↵^⇤is a subsequence of↵then (↵^⇤) = (↵) =x, hence 2 B(x), which proves that that ⌧_x is a winning strategy in G_B(x)

hencex2aB=A. ⇤

3. Non meager a sets in product spaces

We consider now subsets of the product space 2^!⇥X where the first factor is viewed as a parameter space, and denote by ⇡ : 2^!⇥X ! X the canonical projection on X. The main goal of this section is to prove the following result (see next paragraph for the notation):

Theorem 3.1. Let be a Moschovakis class of Borel sets such that the class a has the!-reduction property. Then for any perfect Polish spaceX and any set A⇢2^!⇥X in a there exists a a -measurable function : ⇡⁺(A) ! C0(2^!, X) such that for all

"2⇡⁺(A),Rg( ("))⇢A(").

Notation 3.2. a) For any setA⇢2^!⇥X and any"22^! we denote byA(")⇢X its section inX, and for all open setU inX we define:

⇡_U(A) ={"2 2^!:A(")\U 6=;}

⇡⁺_U(A) ={"2 2^!:A(") is non meager inU}

⇡⁺⁺_U (A) ={"2 2^!:A(") is comeager inU}.

IfU =X we shall write simply: ⇡(A),⇡⁺(A),⇡⁺⁺(A). Note that conversely⇡_U(A),

⇡⁺_U(A),⇡⁺⁺_U (A) are also respectively⇡(AU), ⇡⁺(AU),⇡⁺⁺(AU) whereAU =A\(2^!⇥ U), hence all properties of ⇡(A),⇡⁺(A),⇡⁺⁺(A) can be reformulated as properties of

⇡U(AU),⇡⁺_U(AU),⇡_U⁺⁺(AU).

Note that by the results of Sections 2.2 and 2.3, if A = aB and the game G˜^⇤⇤_B is determined then

"2⇡⁺(A) () Player I wins the gameG˜^⇤⇤_B(")

"2⇡⁺⁺(A) () 8m2!, "2⇡⁺(AUm)

Hence if A 2 a with ⇢ ¹1 then ⇡⁺(A) and ⇡⁺⁺(A) are in a too, while the complexity of⇡(A) might jump up to⌃¹₂.

b) We denote byC(2^!, X) the Polish space of all continuous functions from 2^! toX, equipped with the distance du of uniform convergence. We also denote by C0(2^!, X) the subset of C(2^!, X) constituted of all one-to-one functions. As one can easily check C0(2^!, X) is aG subset ofC(2^!, X) hence a Polish space too.

c) For any functionf :A!B we denote by Rg(f) its rangef(A)⇢B.

d) We recall that (U_m)_m2! is a fixed enumerated basis of nonempty open sets inX. For anys2!^<! of lengthk >0 we sets^⇤=s_{|k 1}andU_s=U_mwhere m=s(k 1), in particularU_hni=U_n; and if moreover s^⇤6=;then we sets^⇤⇤= (s^⇤)^⇤.

The proof of Theorem 3.1 relies on the following structure result which provides, in a computable way, for any setA⇢2^!⇥Xina a subsetGina such that⇡⁺(A) =⇡⁺(G) and each section G(") with "2⇡⁺(G) is a denseG subset of some basic open set U_n"

ofX.

Lemma 3.3. Suppose that is a Moschovakis class. Then for any set A⇢2^!⇥X in a there exists a sequence(Tn)_n2! of a -measurable functions

Tn:⇡⁺(A)!P(!²ⁿ⁺¹)⇡2^(!2n+1) satisfying for all "2⇡⁺(A):

(i) T

n2!

S

t2Tn(")Ut⇢A("),

(ii) T0(") ={⌧0(")}⇢! is a singleton andA(")\U_⌧0(")is comeager in U_⌧0("), and for alln >0:

(iii) ift2Tn(")then t^⇤⇤2Tn 1("),

(iv) ift⁰6=t2Tn(")thent^0⇤6=t^⇤ andU_t\U_t0 =;, (v) ifs2Tn 1(")then the set S

{U_t:t2Tn(")andt^⇤⇤=s}is dense in U_s. Moreover if the Polish spaceX is perfect then for alln >0:

(vi) Tn(")is an infinite subset of!²ⁿ⁺¹.

Proof. SetA=aB withB ⇢2^!⇥X⇥!^! in . Then as observed above:

"2⇡⁺(A) () Player I wins the gameG˜^⇤⇤_B(").

Since is a Moschovakis class we can fix aa -measurable function ˜ :"7!˜_" which assigns to any " 2 ⇡⁺(A) a winning strategy for Player I in the game G˜^⇤⇤_A("). Then composing with the mapping given by Theorem 2.6 we get a a -measurable function : "7! "= (˜_") which assigns to any"2 ⇡⁺(A) a winning strategy for Player I in the gameG^⇤⇤_A(").

We recall thatT denotes the game tree of the Banach-Mazur game. For alln2! let, Sn={(", s)2⇡⁺(A)⇥T :|s|= 2n+ 1 andsis compatible with "}.

We construct Tn satisfying clauses (ii) to (v) by induction onn. Note that S0(") = { "(;)}and since the strategy " is winning for Player I thenA(")\U _"(;) is comeager

inU _"(;). We then defineT0(") ={ "(;)}which satisfies(ii), while the other clauses are empty in this case.

So suppose that Tn is defined satisfying clauses (iii), (iv), (v)and the extra clause

“Tn(") ⇢ Sn(")” and fix an enumeration (t^(j))_j2! of the set T \!²ⁿ⁺³. Given any (", s)2Tn we shall first define a (finite or infinite) increasing sequence (j_k^(",s))k<K(",s)= (jk)k<K in! and then define

Tn+1={(", t^(jk(",s))) :k < K(", s) : (", s)2Tn}

For this, settingj ₁= 1 andU_t( 1) =;, we define by induction the sequence (j_k, J_k)_k<K by:

Jk ={j > jk 1: t^(j)is compatible withsand " andU_t(j)\ [

h<k

U_t(jh) =;}. and

K= min{k2!: Jk 6=;} and 8k < K, jk = minJk

Then by constructionTn+1(")⇢ Sn+1(") and clause (iii) is clearly satisfied at level n+1. Note that a sequencet2!^<!of odd length and compatible with a given strategy is entirely determined byt^⇤hence the first part of clause(iv)is satisfied and it follows from the induction hypothesis and the definition ofTn+1 that the second part of clause(iv)is also satisfied, and we now prove that clause(v)is satisfied. So suppose by contradiction that for some (", s)2Tn the set

V_s:=[

{U_t: t2Tn+1(") andt s}

were not dense inUs. Recall thats= (m0,· · ·, m2n+1) is a finite run in the gameG^⇤⇤_A(") and andU_s=U_m2n+1. SinceV_sis not dense inU_s we can find somemsuch that the set U_m⇢U_s\V_sand diam(U_m) 1

2diam(U_s). Thenmis a legal move for Player II in the game, hencet=s^_hmi^_h "(s^_hmi)i 2T\!²ⁿ⁺³ and ift=t⁽ⁱ⁾then sincet62Tn+1(") there exists someksuch thatj_{k 1}< i < j_k which contradicts the minimality ofj_k.

This ends up the construction of the functionsTnsatisfying(ii)to(v). Note that since the function :"7! "isa -measurable with domain ina then for alls2T the sets

Es:={"2⇡⁺(A) : sis compatible with "}and⇡⁺(A)\Es

are both ina . Then inspecting the definitions above one easily checks by a straightfor- ward induction that the set{"2⇡⁺(A) : s2Tn(")}is ina . HenceTnisa -measurable.

Finally ifx2T

n2!

S

t2Tn(")U_tthen there exists a unique infinite run↵in the game G^⇤⇤_A(") compatible with " such thatx2T

n2!U↵_|n; and since " is a winning strategy for Player I then x2A("), which proves clause(i). This finishes the proof of the first part of Lemma 3.3.

For the last part observe that if moreover the spaceX is perfect then at each level of the previous construction one can replace the setJk by:

J_k⁰ :={j2Jk: Us\⇣

U_t(j)[ [

h<k

U_t(jh)

⌘

6

=;}

which is then nonempty for allk; and it follows thatTn+1(") is infinite. ⇤ Lemma 3.4. Suppose that is a Moschovakis class, a has the!-reduction property, and the Polish space X is perfect. Then for any set A ⇢2^!⇥X in a there exists a

sequence (⌧n)_n2! of a -measurable functions⌧n :⇡⁺(A)⇥2ⁿ!!²ⁿ⁺¹ satisfying for all

"2⇡⁺(A), alln and alls22ⁿ:

(i) ⌧_n(",·) : 2ⁿ!!²ⁿ⁺¹ is one-to-one (ii) ⌧n(", s)2Tn(")

and ifn >0(so that |s| 1and|⌧_n(", s)| 3) then:

(iii) ⌧_n(", s) ^⇤⇤=⌧_{n 1}(", s^⇤)

Proof. We construct⌧_n by induction onn. For n= 0 let ⌧₀:⇡⁺(A)⇥2⁰⇡⇡⁺(A)!! be the function defined in clause(ii)of Lemma 3.3 which satisfies clauses(i)and(ii) of Lemma 3.4.

Suppose now that⌧_n is defined and satisfies clauses(i), (ii), and (iii) ifn >0. Fix

¯t = (t_s)_s22ⁿ 2 (!²ⁿ⁺¹)²ⁿ and identifying the function ⌧_n to a function ˆ⌧_n : ⇡⁺(A) ! (!²ⁿ⁺¹)²ⁿ letP =Pt¯= ˆ⌧_n¹({¯t}). Note that by the induction hypothesis, for any"2P and anys22ⁿ,ts=⌧n(", s)2Tn(") hence by clause(vi)of Lemma 3.3 we can find two elementsus6=vs2Tn+1(") extendingts. Then for any family ¯w= (ws)_s22ⁿ where each w_sis a pair (u_s, v_s) of distinct elements in (!²ⁿ⁺³)²ⁿ⁺¹, let

Q_w¯={"2P : 8s22ⁿ, u_s, v_s2Tn+1(") andu^⇤⇤_s =v^⇤⇤_s =t_s}

Since Tn is a -measurable then each set Q_w¯ is in a and by the observations above P =S

¯

wQw¯. And since the class a has the!-reduction property, we can find a a partition (Q⁰_j)_j2! of P finer than the covering (Qw¯). So for allj there exists a unique

¯

w^j = (u^j_s, v_s^j)_s22ⁿ as above such thatQ⁰_j ⇢Q_w¯^j. We then define for all"2Q⁰_j and all s22ⁿ,

⌧n+1(", s^_0) =u^j_s and ⌧n+1(", s^_1) =v_s^j.

Then variating the parameter ¯tin (!²ⁿ⁺¹)²ⁿone gets a function⌧_n+1:⇡⁺(A)⇥2ⁿ⁺¹!

!²ⁿ⁺³ satisfying clearly all the clauses of Lemma 3.4. ⇤ Proof of Theorem 3.1: Let (Tn)_n2! be as in Lemma 3.3 with the additional clause(vi) since the spaceXis perfect. Observe that given any"2⇡⁺(A), it follows from Lemma 3.3 and Lemma 3.4 that for all 22^! the setT

nU_{⌧n(", |n)} is a singleton{x }ofX, and we define (") to be the function': 7!x . Since for allt2Tn("), diam(U_t)2 ^{2n 1}&0 the function'is continuous, and it follows from clause(i)of Lemma 3.4 and clause(iv) of Lemma 3.3 that'is also one-to-one; hence (")2C0(2^!, X). Moreover by clause(i) of Lemma 3.3 for all 22^!, (", x )2Ahence Rg( ("))⇢A(").

Finally note that if we fix for allt2T some elementat2Utand consider for alln2! and all"2⇡⁺(A) the locally constant function _n^": 2^!!Xdefined by ^"_n(Ns) =a_⌧n(",s) then, since the function⌧_n is a -measurable, the function _n :⇡⁺(A)!C(2^!, X), de- fined by _n(") = ^"_n, isa -measurable too. It follows then from clause(ii)of Lemma 3.4 thatd_u( ("), _n("))2 ²ⁿ. So the function is the uniform limit of a sequence ( _n)_n ofa -measurable functions, hence isa -measurable too. ⇤ Remark 3.5. Note that for alln, k2! the set

E_(n,k)={"2⇡⁺(A) : 9t2Tn("), Ut=Uk} is ina and

G:=\

n

[

k

E_(n,k)⇥U_k ⇢A with⇡(G) =⇡⁺(G) =⇡⁺(A).

Similarly working with strategies for Player II we can prove that under the same assumptions on , ifA⇢2^!⇥X is ina has comeager sections inX then there exists a family (E_(n,k))_(n,k)2!² ofa subsets of 2^! such that the set

G=\

n

[

k

E_(n,k)⇥U_k ⇢A

and all sections ofGare denseG subsets ofX.

4. Proof of Theorem 1.4

The proof of Theorem 1.4 will make use of the following folklore result that we state without proof.

Lemma 4.1. Let ⇤be a!^!-parametrized class with the!-reduction property. Then for any Polish spacesX andY there exists a⇤-measurable function⇥:U !Y with domain U ⇢ 2^!⇥X in ⇤ such that for any ⇤-measurable function ' : D ! Y with domain D⇢X in⇤there exists some ↵22^! such that D=U(↵)and'=⇥(↵,·).

We now start the proof of Theorem 1.4. Note that since any two Polish spaces are Borel isomorphic and the classa is invariant under Borel isomorphisms we may suppose that X = 2^! equipped with some compatible metric. Also since the class a has the

!-reduction property, then applying Lemma 4.1 withX = 2^!,Y =C0:=C0(2^!,2^!)⇡!^! and⇤=a we can fix a universala -measurable function

⇥:U ⇢2^!⇥2^!!C0

as in Lemma 4.1. We then set

={↵22^!: (↵,↵)2U} and for all↵2 :

✓_↵=⇥(↵,↵)2C0 .

Then is ina and the function↵7!✓↵from to C0 isa -measurable.

Lemma 4.2. If(A_j)_j2! is a a covering of 2^!⇥2^! then there exist k2! and ↵2 such that Rg(✓_↵)⇢A_k(↵).

Proof. Applying Theorem 3.1 we can find for all j 2 ! a a -measurable function j :

⇡⁺(A_j)!C0 such that for all"2⇡⁺(A_j), Rg( _j("))⇢A_j("). And for all"22^!, since S

jA_j(") = 2^! then A_j(") is non meager for some j hence (⇡⁺(A_j))_j2! constitutes a covering of 2^!, and again by the!-reduction property ofa we can find aa partition (Pk)_k2! of 2^! and a sequence (jk)_k2! such that for all k, Pk ⇢ ⇡⁺(Ajk). Then = S

k jk|Pk : 2^!!C0 is aa -measurable function, and so there exists some↵such that

= ⇥(↵,·), and since dom ( ) = 2^! = U(↵) then ↵ 2 . Hence (↵) = ✓_↵ and if

↵2P_k then Rg(✓_↵) = Rg( (↵))⇢A_k(↵). ⇤

For alln!, let⇤_n = 2ⁿ⇥ ⁿ⇥(2^!)¹⁺ⁿ and given any

= (i_k)_k<n, ↵_k)_k<n,(⇠_k)_k<1+n 2⇤_n we set:

(0)= (i_k)_k<n , ⁽¹⁾= (↵_k)_k<n ; ⁽²⁾= (⇠_k)_k<1+n . and we shall refer to⇠0 the first coordinate of ⁽²⁾ as theinitial valueof .

We now fix a homeomorphism : 2^! ! 2^!⇥2^!. We shall say that 2 ⇤n is a coherent family of ordernif for all k < n:

(?)_k (⇠_k) = ↵_k,hi_ki^_✓_↵k(⇠_1+k) .

We denote by⌃n the set of all coherent families of ordern, bySn the set of all initial values of such families, and by ⇡n : ⌃n ! Sn the (continuous) function which to any 2⌃_n assigns its initial value. Whenn=! we shall also use the notation (⇡,⌃,S) for (⇡_!,⌃_!,S!).

Lemma 4.3. For alln!,⌃n and Sn are in a , ⇡n :⌃n !Sn is a bijection and its inverse⇡^⇤_n:Sn!⌃n isa -measurable.

Proof. Since and all the ✓_↵ are one-to-one functions it follows by a straightforward induction that a coherent family is entirely determined by its initial value. Hence the continuous⇡_n is one-to-one on⌃_n and⇡_n is a continuous bijection.

Let":C0⇥2^!!2^! denote the evaluation mapping defined by "(f,⇠) =f(⇠). Note that"is continuous and the set

F ={(f,⌘)2C0⇥2^!: ⌘2Rg(f)}={(f,⌘)2C0⇥2^!: 9⇠22^!, ⌘=f(⇠)} is a closed subset ofC0⇥2^! as the projection of a closed subset of (C0⇥2^!)⇥2^! along the compact space 2^!. Then sincea has the substitution property the set

G={(↵,⌘)2 ⇥2^!: ⌘2Rg(✓_↵)}

is in a . It also follows that the inverse evaluation mapping"^⇤ : F ! 2^! defined by

"^⇤(f,⌘) =f ¹(⌘) has a closed graph, hence"^⇤is Borel (of the first Baire class), and the mapping"_⇤:G!2^! defined by"_⇤(↵,⌘) ="^⇤(✓↵,⌘) isa -measurable.

Let 0 : 2^! !2^! and 1: 2^! !2^! denote the coordinate functions of . So for all

⇠ 2 2^!, (⇠) = ( ₀(⇠), ₁(⇠)) and if ₁(⇠) = (i_n)_n2! 2 2^! we set ⁰₁(⇠) = i₀ 2 2 and

001(⇠) = (i_n+1)_n2!22^!, so that ₁(⇠) =h ⁰1(x)i^{_ 00}1(⇠). Then

(?)_k () 8>

<

>:

i_k = ₁⁰(⇠_k)

↵_k = ₀(⇠_k)

⇠1+k ='(⇠k)

where 'is the partial a -measurable function defined by'(⇠) ="_⇤( ₀(⇠), ₁⁰⁰(⇠)) with domainH = ( ₀⇥ ⁰⁰1) ¹(G) ina .

Note that by definition Sn =T

k<n' ^k(H), hence Sn is in a . Note also that the set ⇤^⇤_n of all coherent families is a Borel subset of ⇤_n and if p_n : ⇤_n ! 2^! is the continuous function which to any 2 ⇤n assigns its initial value then by definition

⌃n=⇤^⇤_n\p_n¹(Sn); hence⌃n is ina too. Finally it is clear that the equivalence above provides a a -measurable computation of 2⌃n from its initial value ⇠0 2Sn, hence

⇡^⇤_n:Sn!⌃n isa -measurable. ⇤

Lemma 4.4. For anya -measurable functionf : 2^!⇥2^!!2^! there exist↵_⇤22^! and a continuous functiong: 2^!!2^! such that:

8x22^!,9 2⌃: ⁽⁰⁾ =x and g(x) =f(↵_⇤,✓_↵⇤(⇡( ))) Proof. For alln2!letfn : 2^!⇥2^!!2 denote then^th coordinate off.

Let fork22,Ak={(↵,⌘)22^!⇥2^!: f0(↵,⌘) =k}. Sincef0isa -measurable then (A0, A1) is a a -partition of 2^!⇥2^! and applying Lemma 4.2 we can fix ↵_⇤ 2 and k_⇤22 such that for all⇠22^!,f0(↵_⇤,✓↵_⇤(⇠)) =k_⇤.

Let :S

n2!2ⁿ⇥ ⁿ⇥2^!!2^!be the mapping defined inductively by:

( (;,;,⇠) =⇠

(¯ı^_hji,↵¯^_h i,⇠) = (¯ı,↵,¯ ¹( ,hji^_✓ (⇠))

Claim 4.5. For all (¯ı,↵)¯ 2 2ⁿ⇥ ⁿ, there exists ⇠¯= (⇠_m)_mn 2 (2^!)ⁿ⁺¹ such that (¯ı,↵,¯ ⇠)¯ 2⌃_n and for allmn, (¯ı_|m,↵¯_|m,⇠_m)is constant.

Proof: Set ¯ı = (i_m)_m<n, ¯↵= (↵_m)_m<n, and let ¯⇠ = (⇠_m)_mn 2 (2^!)ⁿ⁺¹ defined by a descending induction by choosing⇠_n arbitrary in 2^!and form < n,

⇠_m= ¹(↵_m,hi_mi^_✓_↵m(⇠_m+1))

so that (¯ı,↵,¯ ⇠)¯ 2⌃_n. Then by definition of we have for allmn,

(¯↵_|m+1,¯ı_|m+1,⇠_m+1) = (¯↵_|m,¯ı_|m, ¹(↵_m,hi_mi^_✓_↵m(⇠_m+1))) = (¯↵_|m,¯ı_|m,⇠_m). ⇧ Claim 4.6. There exists a function 2^<!3s7!(↵s, ks)2 ⇥2satisfying for all s22ⁿ and all⇠22^!,f_n ↵_⇤,✓_↵⇤( (s,↵_s,⇠)) =k_s where↵_s= (↵_s|m)_m<|s|.

Proof: Fors=;we define (↵_;, k_;) = (↵_⇤, k_⇤) which satisfies the claim. So suppose that

↵=↵_s⇤2 ⁿ is already defined, and let s22ⁿ⁺¹. Sincea has the weak substitution property andf_n+1 is a totala -measurable function then the sets

Bk={( ,⇠)22^!⇥2^!: fn+1(↵_⇤,✓↵_⇤( (s,↵^_h i,⇠)) =k}

fork22 form aa -partition of 2^!⇥2^!, and applying Lemma 4.2 we can find (↵_s, k_s)2

⇥2 such that Rg(✓_↵s)⇢B_ks(↵_s), hencef_n+1 ↵_⇤,✓_↵⇤( (↵_s,⇠)) =k_sfor all⇠22^!. ⇧ We fix now a function s7! (↵s, ks) as in Claim 4.6 and denote by g : 2^! ! 2^! and h: 2^! ! ^! the continuous functions defined for allx 22^! by g(x) = (kx_|n)_n2! and h(x) = (↵x_|n)_n2!. Also for alln2!we denote bygn : 2^!!2 the n^th coordinate ofg.

Claim 4.7. For alls22ⁿ, there exists 2⌃_n such that ⁽⁰⁾=sand for allmnand allx s,g_m(x) =f_m(↵_⇤,✓_↵⇤(⇡( ))).

Proof: Givens22ⁿ let 2⌃_n be given by Claim 4.5 applied to (¯ı,↵) = (s,¯ ↵_s). Then by definition ofg for allx sand allmn,g_m(x) =k_s|m hence by Claim 4.6:

gm(x) =fm ↵_⇤,✓↵_⇤( (s_|m,↵s_|m,⇠m))

=fm ↵_⇤,✓↵_⇤( (s_{|m 1},↵s_|m 1,⇠m 1))

=· · ·

=fm ↵_⇤,✓↵_⇤( (;,;,⇠0)) .

and (;,;,⇠₀) =⇠₀=⇡( ). ⇧

Fixx22^!; for alln2!set sn=x_|n and let

K_n(x) ={ 2⌃_n: ( ⁽⁰⁾, ⁽¹⁾) = (s_n,↵_sn) and 8mn, g_m(x) =f_m(↵_⇤,✓_↵⇤(⇡( )))}

which is clearly a compact subset of {(sn,↵sn)}⇥(2^!)ⁿ⁺¹. Note that if for all mn,

⇡^m_n : Kn(x) ! Km(x) denotes the canonical restriction operator defined by ⇡_n^m =

( _|m⁽⁰⁾, _|m⁽¹⁾, ⁽²⁾_|m+1) then for all k  m  n, ⇡_n^k = ⇡^k_m ⇡^m_n. Hence (K_n(x),⇡^m_n)_mn is a projective system and we can consider its inverse limit which can clearly be identified to the compact set

K(x) ={ 2⌃: ( ⁽⁰⁾, ⁽¹⁾) = (x, h(x)), and8n, gn(x) =fn(↵_⇤,✓↵_⇤(⇡( )))}. Since by Claim 4.7 each compact setKn(x) is nonempty thenK(x) is nonempty too (see [1], ch.1,§9, Sect. 6, Prop. 8), which finishes the proof of Lemma 4.4. ⇤ End of the proof of Theorem 1.4: Suppose thatA⇢2^!is weaklya -hard. To show that Aisa -hard consider any setB⇢2^! ina and let

Bˆ ={(↵,⌘)2 ⇥2^!: 9 2⌃, ⌘=✓↵(⇡( )) and ⁽⁰⁾2B}

={(↵,⌘)2G: ✓_↵¹(⌘)2S, and⇡^⇤(✓_↵¹(⌘))2 ^!⇥B⇥(2^!)^!}.

Then the set ˆB is ina and sinceA is is weaklya -hard there exists a a -measurable functionf : 2^!⇥2^! !2^! such ˆB =f ¹(A). Then for all x22^! if 2⌃ is given by Lemma 4.4 then:

g(x) =f(↵_⇤,✓↵_⇤(⇡( )))2A () (↵_⇤,✓↵^⇤(⇡( )))2Bˆ () x= ⁽⁰⁾2B hencegis a continuous reduction ofB toA. This proves thatAisa -hard . ⇤

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Gabriel Debs, Sorbonne Universit´e, Universit´e Paris Diderot, CNRS, Institut de Math´ema- tiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France, and Universit´e Le Havre Normandie, Institut Universitaire de Technologie, Rue Boris Vian, BP 4006 76610 Le Havre, France.

E-mail address:gabriel.debs@imj-prg.fr

Jean Saint Raymond, Sorbonne Universit´e, Universit´e Paris Diderot, CNRS, Institut de Math´ematiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005, Paris, France

E-mail address:jean.saint-raymond@imj-prg.fr