Topology on the space of weak equivalence classes

TOPOLOGY AND CONVEXITY IN THE SPACE OF ACTIONS MODULO WEAK EQUIVALENCE

2.3 Topology on the space of weak equivalence classes

Let Γ be a countable group and A∼(Γ,X, µ) be its space of actions modulo weak equivalence. We consider a metric on A∼(Γ,X, µ)which is implicit in [1].

Fix an enumeration (γi)^∞

i=0 of Γ. If A = {A₁, . . . ,Ak} is a partition of X into k pieces, a ∈ A(Γ,X, µ) and n ∈ N, let M_n,k^A(a) ∈ [0,1]^n×k×k be the point whose p,q,r coordinate is µ(γ_p^aAq ∩ Ar), where p ≤ n and q,r ≤ k. Let C_n,k(a) = {M_n,k^A(a):Ais a partition ofX intok pieces.} Then we can define a pseudometricdon A(Γ,X, µ)by the formula

d(a,b)=

∞

Õ

n,k=1

1

2^n+kdH(C_n,k(a),C_n,k(b)),

where dH is the Hausdorff distance in the hyperspace of compact subsets of [0,1]^n×k×k. It is easy to see thata∼ bif and only ifd(a,b)= 0, soddescends to a metric on A∼(Γ,X, µ), which we also denote by d. Letτ₁ be the topology induced by d. We note that this definition extends to actions on countable spaces. We will write A^∗_∼(Γ)for the space of all actions ofΓon probability spaces.

We now describe a different construction of the topology on A∼(Γ,X, µ) due to Tucker-Drob [74] in order to show it agrees with the one we have just introduced.

(Tucker-Drob shows in [74] that his formulation agrees with the one from [1]).

Let S be a compact Polish space, and consider S^Γ, which is also a compact Pol-

ish space. Γ acts on S^Γ by the shift action s given by (γ^sf)(δ) = f(γ⁻¹δ). Let Ms(S^Γ) be the compact Polish space of shift-invariant probability measures on S^Γ and letK_S =K(Ms(S^Γ))be the hyperspace of compact subsets of Ms(S^Γ)equipped with the Hausdorff topology. Then K_S is again compact and Polish. Now con- sider anS-valued random variable φ ∈ L(X, µ,S)onX, that is to say a measurable mapφ : X → S. For each measure-preserving actiona ∈ A(Γ,X, µ)we get a map Φ^φ,a_S : X → S^Γby lettingΦ^φ,a_S (x)(γ)= φ((γ⁻¹)^ax)and consequently a shift-invariant measure(Φ^φ,a

S )_∗µonS^Γ. Then define a subsetE(a,S)ofMs(S^Γ)by E(a,S)= {(Φ^φ,a

S )_∗µ:φ: X → Sis measurable}.

LetΦ_S : A(Γ,X, µ) → K_Sbe given byΦ_S(a)= E(a,S). WhenS =K is the Cantor set, we omit the subscript S on the notations just introduced. By Proposition 3.5 in [74], we have a ∼ bif and only if Φ(a) = Φ(b) so we can consider the initial topology on A∼(Γ,X, µ)induced byΦ. Call thisτ₂. We now work towards showing τ₁agrees withτ₂. There will be a series of preliminary steps. This entire argument can be regarded as a ‘perturbed’ version of Proposition 3.5 in [74].

We first fix a compatible metric on Ms(K^Γ). Let A_K be the collection of clopen subsets ofK^Γ of the formπ_F⁻¹

Îγ∈F Aγ

whereAγ ⊆ K an element of some fixed countable clopen basis for K, F ⊆ Γ is finite and π : K^Γ → K^F is the projection onto the F-coordinates. Since the elements ofA_K generate the Borelσ-algebra of K^Γ, for(νn)^∞_n=

1 ⊆ Ms(K^Γ)we haveνn → νin Ms(K^Γ)if and only ifνn(A) → ν(A) for every A∈ A_K. So, enumerating the elements ofA_K as(A_i^K)^∞_i=1,δK given by

δK(ν, ρ)=

∞

Õ

i=1

1

2ⁱ|ν(A_i^K) − ρ(A_i^K)|

is a compatible metric onMs(K^Γ).

Lemma 2.3.1. For any > 0 there exists k ∈ N such that every a and every φ ∈ L(X, µ,K) there is ψ ∈ L(X, µ,K) with δK((Φ^φ,a)_∗µ,(Φ^ψ,a)_∗µ) < such that the range ofψhas size≤ k. Note that kdepends only on, not onaorφ.

Proof. Fix . Choose N large enough that Í∞ i=N 1

2ⁱ < . For each i ≤ N, write Ai = π⁻_F¹

i

Îγ∈Fi Aⁱ_γ

for Aⁱ_γ ⊆ K clopen and Fi ⊆ Γ finite. We have for all

a ∈A(Γ,X, µ)andφ, ψ ∈ L(X, µ,K),

|Φ^φ,a(Ai) −Φ^ψ,a(Ai)| =

Φ^φ,a π⁻¹_Fi Ö

γ∈F_i

Aⁱ_γ

! !

−Φ^ψ,a π_F⁻¹_i Ö

γ∈F_i

Aⁱ_γ

! !

= |µ({x:Φ^φ,a(x)(γ) ∈ Aⁱ_γ for allγ ∈ Fi})

− µ({x:Φ^ψ,a(x)(γ) ∈ Aⁱ_γ for allγ ∈ Fi})|

= |µ({x: φ((γ⁻¹)^ax) ∈ Aⁱ_γfor allγ ∈ Fi})

− µ({x:ψ((γ⁻¹)^ax) ∈ Aⁱ_γ for allγ ∈Fi})|

=

µ Ù

γ∈Fi

γ^aφ⁻¹(Aⁱ_γ)

!

− Ù

γ∈Fi

γ^aψ⁻¹(Aⁱ_γ)

!

. (2.1)

Now, fixφ∈ L(X, µ,K). Let(Bj)^k_j=

1be the finite partition ofK given by the atoms of the Boolean algebra generated by (Aⁱ_γ)_{i≤N,γ∈Fi}. Note that k depends only on . For each j ≤ k, let yj be any point in Bj. Define a mapψ : X → K by letting ψ(x)= yj for the unique jsuch thatx ∈ φ⁻¹(Bj). Thenψ⁻¹(Bj)= φ⁻¹(Bj)for each j, and hence φ⁻¹(Aⁱ_γ)=ψ⁻¹(Aⁱ_γ)for eachi ≤ N andγ ∈Fi. Therefore the value of the expression(1)is 0 andδK((Φ^φ,a)_∗µ,(Φ^ψ,a)_∗µ)< . Lemma 2.3.2. If E(an,L) → E(a,L) in K(Ms(L^Γ)) for every finite set L then E(an,K) → E(a,K)inK(Ms(K^Γ)).

Proof. Fix > 0 in order to show that eventuallydK

E(an,K),E(a,K)

< , where dK is the Hausdorff distance in K(Ms(K^Γ)) constructed from δK. For k ∈ N and b ∈A(Γ,X, µ)let

Ek(b,K)= {(Φ^φ,a)_∗µ: φ:X →K is measurable and the range ofφhas size ≤ k}.

By Lemma 2.3.1 we can choosek ∈ Nsuch that E(b,K) ⊆ B

4(Ek(b,K))for every b ∈A(Γ,X, µ)whereBr(A)={ν ∈ Ms(K^Γ):δK(ν, ρ)< rfor some ρ∈ A}. Notice thatEk(b,K)=Ð

L⊆K,

|L|=k

E(b,L). Fix a setLof sizekand chooseNlarge enough such that ifn≥ N thendK_L

E(an,L),E(a,L)

< ₄ wheredK_L is the Hausdorff distance inK(Ms(L^Γ)). Since the construction is independent of the set chosen to realize L, we have in fact dK_L

E(an,L),E(a,L)

< ₄ for every finite set L of size k. For a fixed finite L ⊆ K letEL(b,K) = {(Φ^b,φ)_∗µ : φ : X → K measurable,φ(X) ⊆ L}. Then for anyb,c∈A(Γ,X, µ)we have

dK

EL(b,K),EL(c,K)

= dK_L

E(b,L),E(c,L) ,

so that whenn ≥ N,

dK

Ek(an,K),Ek(a,K)

=dK

©

­

­

­

« Ø

L⊆K

|L|=k

E(an,L), Ø

L⊆K

|L|=k

E(a,L) ª

®

®

®

¬

≤ sup

L⊆K

|L|=k

dK_L

E(an,L),E(a,L)

<

4. Therefore whenn≥ N,

dK

E(an,K),E(a,K)

≤ dK

E(an,K),Ek(an,K) +dK

Ek(an,K),Ek(a,K) +dK

Ek(a,K),E(a,K)

< 3 4 .

Lemma 2.3.3. LetLbe a finite set of sizek. Then for each finite set(Ap)^q

p=1of basic clopen setsAp⊆ L^Γ and > 0there isδ >0such that ifd(a,b)< δthen for allφ∈ L(X, µ,L)there existsψ ∈ L(X, µ,L)such that |(Φ^a,φ_L )_∗µ(Ap) − (Φ^b_L^,ψ)_∗µ(Ap)| <

for allp ≤ q.

Proof. Write Ap = Ñ

γ∈F_pπ⁻_γ¹(p(γ))for some Fp ⊆ Γ finite,  : Fp → k and fix > 0. Choose a finite F ⊆ Γ with(Fp)² ⊆ F for all p ≤ q. We may assume the identitye∈ F. Supposed(a,b) < ^δ

2^|F|+k|F|; we will specify a value forδ later. Now fixφ : X → k and letBi = φ⁻¹(i). Givenη : F → k, letBη = Ñ

γ∈Fγ^aB_η(γ). We can then find a partition{Dη}_η∈kF such that

|µ(γ^aBη₁ ∩Bη₂) −µ(γ^bDη₁∩Dη₂)| < δ

for all η1, η2 ∈ k^F and γ ∈ F. Define ψ : X → k, by ψ(y) = l if y ∈ Dη for someη with η(e) = l. Furthermore, for eachl ≤ k let Dl = Ã{Dη : η ∈ k^F and η(e) = l} = ψ⁻¹(l). For each J ⊆ F and σ ∈ k^J let D_σ = Ã{D_η : η ∈ k^F and σ v η}, whereσ v ηmeansηextendsσand let ˜Dσ =Ñ

γ∈Jγ^bD_σ(γ). Furthermore ifγ ∈ Γ, J ⊆ Γandσ ∈ k^J letγ ·σ ∈ k^γJ be given by(γ ·σ)(δ)= σ(γ⁻¹δ). For σ ∈ K^Fp andγ ∈Fpwe have

|µ(γ^bDσ∩Dγ·σ) −µ(γ^aBσ∩Bγ·σ)|

≤ Õ

(η∈k^F:σvη)

Õ

(η⁰∈k^F:γ·σvη⁰)

|µ(γ^bDη∩Dη⁰) −µ(γ^aBη∩Bη⁰)|

≤ δ(k^|F|)².

In particular, settingγ =ewe see|µ(Bσ) −µ(Dσ)| < δk^2|F| for everyσ : Fp→ k. Sinceγ^aBσ = Bγ·σ = γ^aBσ∩Bγ·σ we have

|µ(Dσ) −µ(γ^bDσ∩Dγ·σ)| ≤ |µ(Dσ) − µ(γ^aBσ)|

+|µ(γ^aBσ∩Bγ·σ) − µ(γ^bDσ∩Dγ·σ)|

= |µ(Dσ) − µ(Bσ)|

+|µ(γ^aB_σ∩B_γ·σ) − µ(γ^bD_σ∩D_γ·σ)|

<2δk^2|F|

and also

|µ(Dγ·σ) − µ(γ^bDσ∩Dγ·σ)| ≤ |µ(Dγ·σ) − µ(Bγ·σ)|

+|µ(γ^aBσ ∩Bγ·σ) − µ(γ^bDσ∩Dγ·σ)|

< 2δk^2|F|. Therefore

µ((γ^bDσ)4(Dγ·σ))= µ(γ^bDσ)+ µ(Dγ·σ) −2µ(γ^bDσ∩Dγ·σ)

≤ |µ(Dγ·σ) −µ(γ^bDσ∩Dγ·σ)|+|µ(Dγ·σ) − µ(γ^bDσ∩Dγ·σ)|

< 4δk^2|F|. (2.2)

Since(Dη)_η∈kF is a partition ofX and(Fp)² ⊆ F, we have

D_p = Ä

η∈k^F

_pvη

Dη = Ù

γ∈F_p

Ä

σ∈k^γFp σ(γ)=_p(γ)

Dσ = Ù

γ∈F_p

Ä

σ∈k^Fp σ(e)=_p(γ)

Dγ·σ.

Now, by(2),

µ

©

­

­

­

­

«

©

­

­

­

« Ù

γ∈F_p

Ä

σ∈k^Fp σ(e)=(γ)

Dγ·σ

ª

®

®

®

¬ 4

©

­

­

­

« Ù

γ∈F_p

Ä

σ∈k^Fp σ(e)=_p(γ)

γ^bDσ

ª

®

®

®

®

¬ ª

®

®

®

®

¬

< (|Fp|k^|Fp|)(4δk^2|F|). (2.3)

Note that

Ù

γ∈Fp

Ä

σ∈k^Fp σ(e)=_p(γ)

γ^bD_σ = Ù

γ∈Fp

γ^bD_p(γ) = D˜_p,

so(3)reads

|µ(D_p) − µ(D˜_p)| < (|Fp|k^|Fp|)(4δk^2|F|).

Moreover,

(Φ^b,ψ_L )_∗µ(Ap)= µ({x:Φ^b_L^,ψ(x) ∈ Ap})

= µ({x:Φ^b_L^,ψ(x)(γ)= p(γ)for allγ ∈Fp})

= µ({x:ψ((γ⁻¹)^bx)= p(γ)for allγ ∈Fp})

= µ({x: x ∈γ^bψ⁻¹(p(γ))for allγ ∈Fp})

= µ©

­

« Ù

γ∈Fp

γ^bD_p(γ)ª

®

¬

= µ(D˜_p). Similarly,(Φ^a,φ

L )_∗µ(Ap)= µ(B_p). So we finally have

|(Φ^b,ψ_L )_∗µ(Ap) − (Φ^a,φ_L )_∗µ(Ap)| = |µ(D˜_p) −µ(B_p)|

≤ |µ(D˜_p) −µ(D_p)|+|µ(D_p) − µ(B_p)|

<(|Fp|k^|Fp|)(4δk^2|F|)+2δk^2|F|. Since k is fixed in advance, |Fp| ≤ |F| and F depends only on (Ap)^q_p=

1, it is clear thatδcan be chosen so(|Fp|k^|Fp|)(4δk^2|F|)+2δk^2|F| < for allp ≤ q.

We can now prove the main result of this section.

Theorem 2.3.1. τ1=τ2.

Proof. Suppose thatan→ ainτ₁. We need to proveΦ(an) →Φ(a)inK(Ms(K^Γ)).

By Lemma 2.3.2 it suffices to fix a finite set L and show E(an,L) → E(a,L) in K(Ms(L^Γ)). Letk = |L|. WriteEn= E(an,L)andE = E(a,L). As before, if we let A_L = (A_i^L)^∞

i=1 be the collection of clopen subsets of L^Γ of the formÑ

γ∈Fπ_γ⁻¹(jγ) for a finiteF ⊆ Γand jγ ≤ k, then

δL(ν, ρ)=

∞

Õ

i=1

1

2ⁱ|ν(A_i^L) − ρ(A^L_i)|

is a compatible metric on Ms(L^Γ). Fix > 0 in order to show that eventually dL(En,E) < , wheredL is the Hausdorff distance inK(Ms(L^Γ))constructed from δL. ChooseN sufficiently large thatÍ∞

i=N 1

2ⁱ < ₂. By Lemma 2.3.3 there isδ > 0 such that if d(a,b) < δ then for each i ≤ N and all φ ∈ L(X, µ,L) there exists ψ ∈ L(X, µ,L) such that |(Φ^a,φ

L )_∗µ(A_i^L) − (Φ^b,ψ

L )_∗µ(A_i^L)| < ₂. Thus if M is large

enough thatd(an,a)< δforn ≥ M, we havedL(En,E)< .

Now suppose Φ(an) → Φ(a) inK(Ms(K^Γ)). Fixr,q and > 0 in order to show that eventuallydH(Cr,q(an),Cr,q(a)) < . Chooseqdistinct points(xp)^q_p=

1 ∈ K and let(Dp)^q_p=

1 be a family of disjoint clopen subsets of K with xp ∈ Dp. Now let M be large enough that all sets of the form π_γ⁻¹_s(Dp) ∩π_e⁻¹(Dt)for s ≤ r and p,t ≤ q appear as some A^K_i for i ≤ M in our previously chosen clopen basis A_K. Then choose N large enough that when n ≥ N, dK(Φ(an),Φ(a)) < _2M. Then for each φ ∈ L(X, µ,K)we haveψ ∈ L(X, µ,K)such thatδK((Φ^an,φ)_∗µ,(Φ^a,ψ)_∗µ)< _2M. So in particular, ifn≥ Nthen for eachφ ∈L(X, µ,K)there existsψ ∈ L(X, µ,K)such that

|(Φ^an,φ)_∗µ(π_γ⁻¹_s(Dp) ∩π⁻¹_e (Dt)) − (Φ^a,ψ)_∗µ(π⁻¹_γs(Dp) ∩π_e⁻¹(Dt))| <

for allp,t ≤ qands ≤r.

Now suppose n ≥ N and let (Bp)^q_p=1 be a partition of X. Define φ : X → K by takingφ(x)= xpfor the uniquep ≤ qwith x ∈Bpso by the previous paragraph we have a correspondingψ. Observe that for allγ ∈Γwe have

µ(γ^anBp∩Bt)= µ(γ^anφ⁻¹(Dp) ∩φ⁻¹(Dt))

= µ({x: φ((γ^an)⁻¹x) ∈ Dpandφ(x) ∈ Dt})

= µ({x:Φ^φ,an(x)(γ) ∈ DpandΦ^φ,an(x)(e) ∈ Dt})

= µ({x:Φ^φ,an(x) ∈π⁻_γ¹(Dp)andΦ^φ,an(x) ∈π_e⁻¹(Dt)})

= µ({x:Φ^φ,an(x) ∈π⁻¹_γ (Dp) ∩π₁⁻¹(Dt)})

= (Φ^φ,an)_∗µ(π⁻¹_γ (Dp) ∩π₁⁻¹(Dt)). Similarly lettingHp =ψ⁻¹(Dp)we have

µ(γ^aHp∩Ht)= (Φ^ψ,an)_∗µ(π⁻¹_γ (Dp) ∩π₁⁻¹(Dt)). Thus for allp,t ≤ qands ≤ r,

|µ(γ^a_sⁿBp∩Bt) − µ(γ_s^aHp∩Ht)|

= |(Φ^φ,an)_∗µ(π⁻¹_γ+s(Dp) ∩π_e⁻¹(Dt)) − (Φ^ψ,an)_∗µ(π⁻¹_γs(Dp) ∩π_e⁻¹(Dt))|

< .

We have shown that when n ≥ N, Cr.q(an) ⊆ B(Cr,q(a)). The argument that eventuallyCr,q(a) ⊆ B(Cr,q(an))is identical.

Topology on the space of stable weak equivalence classes

Let A∼_s(Γ,X, µ) be the space of stable weak equivalence classes and let ι be the trivial action of Γ on an standard probability space. By Lemma 3.7 in [74], we have a ≺_s b if and only if a ≺ ι× b. Moreover, Theorem 1.1 in [74] says that E(a×ι,K) = cch(E(a,K)), where Ms(K^Γ) carries its natural topological convex structure as a compact convex subset of a Banach space. LettingΨ : A(Γ,X, µ) → K(Ms(K^Γ)) be the map a 7→ cch(E(a,K)) we have Ψ(a) = Ψ(b) if and only if a∼_s b. Tucker-Drob gives A∼_s(Γ,X, µ)the initial topology induced byΨ, in which it is a compact Polish space. Thus we havean →ain the topology of A∼_s(Γ,X, µ)if and only ifan×ι→ a×ιin the topology of A∼(Γ,X, µ). Therefore we can introduce a metricdson A∼_s(Γ,X, µ)by settingds(a,b)= d(a×ι,b×ι).