The space of weak equivalence classes as a weak convex space

TOPOLOGY AND CONVEXITY IN THE SPACE OF ACTIONS MODULO WEAK EQUIVALENCE

2.4 The space of weak equivalence classes as a weak convex space

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).

Choose J₁ so that if j > J₁ then |tj −t| < ₆. Choose J₂ > J₁ so if j > J₂ then dH(Ca_j,Ca) < ₆ anddH(Cb_j,Cb) < ₆. Fix j > J2. Writingθ for the isomorphism from(Y1tY2,tµ+(1−t)µ)to(Y, µ)andθjfor the isomorphism from(Y1tY2,tjµ+ (1−tj)µ)to(Y, µ)we have a partition(Bs,i)^l_s₌

1ofYigiven byBs,i = θ⁻¹(Bs) ∩Yi. So we can find a partition(Ds,i)^l_s₌₁ofYisuch that for allp ≤ mand alls,t ≤ l we have

|µ(γ^a_p^jDs,1∩Dt,1) −µ(γ^a_pBs,1∩Bt,1)| <

6 and

|µ(γ_p^b^jDs,2∩Dt,2) −µ(γ^b_pBs,2∩Bt,2)| <

Now, letDs = θj(Ds,1tDs,2). Note that since eachθj(Yi)iscj invariant, µ(γ^c_p^jDs∩Dt)= µ(γ^c_p^jθj(D_s,1) ∩θj(D_t,1))+µ(γ^c_p^jθj(D_s,2) ∩θj(D_t,2))

= µ(θj(γp^a^jDs,1∩Dt,1))+ µ(θj(γ^bp^jDs,2∩Dt,2))

= tjµ(γ_p^a^jDs,1∩Dt,1)+(1−tj)µ(γ^b_p^jDs,2∩Dt,2). Similarly sinceθ(Yi)isc-invariant we have

µ(γ^c_pBs∩Bt)= µ(γ^c_pθ(Bs,1) ∩θ(Bt,1))+ µ(γ_p^cθ(Bs,2) ∩θ(Bt,2))

= µ(θ(γ^a_pBs,1∩Bt,1))+µ(θ(γ_p^bBs,2∩Bt,2))

= tµ(γ_p^a∩Bs,1∩Bt,1)+(1−t)µ(γ^b_p∩Bs,2∩Bt,2).

|tjµ(γ^a_p^jDs,1∩Dt,1) −tµ(γ^a_p∩Bs,1∩Bt,1)| <

2 and

|(1−tj)µ(γp^b^jDs,2∩Dt,2) − (1−t)µ(γ^b_p∩Bs,2∩Bt,2)| <

2, and hence

|µ(γ^c_p^jDs∩Dt) −µ(γ^c_pBs∩Bt)| <

as claimed.

Now we must show that for sufficiently large j, every partition B₁, . . . ,Bl of Y there is a partition D₁, . . . ,Dl ofY depending on j such that for all s,t ≤ l and p≤ mwe have

|µ(γ_p^cDs∩Dt) − µ(γ_p^c^jBs∩Bt)| < .

The argument is similar to the previous step, so we omit it.

Corollary 2.4.1. A∼(Γ,Y, µ)is path connected.

Corollary 2.4.2. A∼(Γ,Y, µ)is uncountable.

We now record a lemma which will be useful later, guaranteeing that the metric on A∼(Γ,X, µ)behaves nicely with respect to the convex structure.

Lemma 2.4.1. For any convex setK ⊆ A∼(Γ,X, µ)the functiond(·,K)=infb∈Kd(·,b) is convex.

Proof. Letx,y ∈A∼(Γ,X, µ)and considert x+(1−t)y. Fixn,kand writeC(a)for C_n,k(a). It suffices to show that

b∈Kinf dH(C(t x+(1−t)y),C(b)) ≤t(inf

b∈KdH(C(x),C(b))+(1−t)(inf

b∈KdH(C(y),C(b))), wheredH is the Hausdorff distance in the space[0,1]^n×k². Fix > 0. It suffices to finda∈K with

dH(C(t x+(1−t)y),C(a)) ≤t(inf

b∈KdH(C(x),C(b))+)+(1−t)(inf

b∈KdH(C(y),C(b))+). (2.4) Choosec ∈ K with dH(C(x),C(c)) < infb∈KdH(C(x),C(b))+ and choose d ∈ K with dH(C(x),C(d)) < infb∈KdH(C(y),C(b))+ . Note that since K is convex, tc+(1−t)d ∈K. We claim

dH(C(t x+(1−t)y),C(tc+(1−t)d)) ≤tdH(C(x),C(c))+(1−t)dH(C(y),C(d)), which implies(4). Letδ >0, it then suffices to show

dH(C(t x+(1−t)y),C(tc+(1−t)d)) ≤t(dH(C(x),C(c))+δ)+(1−t)(dH(C(y),C(d))+δ).

(2.5) Let X₁ and X₂ be two copies of X and ν be the measure on X₁ t X₂ given by t(µ X₁)+(1−t)(µ X₂). LetP =(Pi)^k

i=1be a partition ofX₁tX₂. This induces a partitionP₁=(P_i¹)_i^k₌

1ofX1given byP_i¹ = Pi∩X1and similarly we have a partition P₂=(P_i²)_i^k₌

1ofX2. We can find a partitionQ₁ =(Q¹_i)_i^k₌

1ofX1such that form ≤ n andi,j ≤ kwe have

|µ(γ_m^xP_i¹∩P¹_j) −µ(γ_m^cQ¹_i ∩Q¹_j)| < dH(C(x),C(c))+δ and similarly we can find a partitionQ₂ = (Q²_i)_i^k₌

1 of X2 such that for m ≤ nand i, j ≤ k we have

|µ(γ_m^yP_i²∩P²_j) − µ(γ_m^dQ_i²∩Q²_j)| < dH(C(y),C(d))+δ.

Let Q = (Qi)^k

i=1 be the partition of X1 t X2 given by Qi = Q¹_i tQ²_i. Write t(dH(C(x),C(c))+δ)+(1−t)(dH(C(y),C(d))+δ) = r. Then for all m ≤ n and i, j ≤ k we have

|ν(γ^{t x}_m⁺^(1−t)yPi∩Pj) −ν(γ_m^tc⁺^(1−t)dQi∩Qj)|

≤ |tµ(γ_m^xP_i¹∩P¹_j) −tµ(γ_m^cQ¹_i ∩Q¹_j)|

+|(1−t)µ(γm^yP_i²∩P²_j) − (1−t)µ(γ_m^dQ²_i ∩Q²_j)|

≤r.

We have shown thatC(t x +(1−t)y) ⊆ Br(C(tc+(1−t)d)). The argument that C(tc+(1−t)d) ⊆ Br(C(t x+(1−t)y))is identical, so we omit it. Thus we conclude dH(C(t x+(1−t)y),C(tc+(1−t)d)) ≤rand(5)holds.

We note that A∼(Γ,X, µ) in fact has additional structure in that it admits convex combinations of infinitely many elements. We first consider the case of a countable convex combination. If λi ∈ [0,1] are such that Í∞

i=1λi = 1 and ai ∈ A∼(Γ,X, µ) then we can naturally define an action Í∞

i=1λiai on the disjoint sum Ã∞

i=1Xi with theicopy of X weighted byλi. It remains to check that this is independent of the choice of representativesai.

Proposition 2.4.2. Ifai ≺ bi for alli, thenÍ∞

i=1λiai ≺ Í∞ i=1λibi. Proof. Let A₁, . . . ,Ak ⊆ Ã∞

m=1Xm, > 0 and F ⊆ Γ finite be given. Choose N such thatÍ∞

m=Nλm < ₂. For eachm < N, consider the partition A₁^m, . . . ,A^m_k ofXm

given by A_i^m = Ai∩Xm. We can find for eachm < N a partition B₁^m, . . . ,B^m_k such that for allγ ∈ Fandi,j ≤ k we have

|µ(γ^aⁱA_i^m∩A^m_j) − µ(γ^bⁱB_i^m∩B^m_j)| <

LetBi = Ã∞

m=1B_i^m. Then

γ^Í^∞^m=1^λ^m^a^mAi∩Aj

− µ

γ^Í^∞^m=1^λ^m^b^mBi∩Bj

≤ |

m=1

λmµ(γ^a^mA^m_i ∩ A^m_j) −

m=1

λmµ(γ^b^mB_i^m∩B^m_j)|

∞

m=M

λmµ(γ^a^mA^m_i ∩ A^m_j) −

∞

m=M

λmµ(γ^b^mB_i^m∩B^m_j)|

≤

m=1

λm|µ(γ^aⁱA^m_i ∩A^m_j) −µ(γ^bⁱB_i^m ∩B^m_j )|+ 2

≤ 2

m=1

λm

! +

2 ≤ .

It is in fact possible to define integrals of weak equivalence classes of actions over a probability measure. Let(Z, η)be a probability space and suppose that for eachzwe have a probability space(Yz, νz)and a measure-preserving actionΓ y^a^z (Yz, νz)such that the map z 7→ [az]from(Z, η)to A^∗_∼(Γ)is measurable, where[az]is the weak equivalence class ofaz. Note that we do not require(Xz, νz)or(Z, η)to be standard.

LetY = Ã

z∈ZYz and put a measure ν onY by taking ν(A) = ∫

Z νz(A∩YZ)dη(z). Y will be a standard probability space isomorphic to (X, µ)if(Z, η)is standard or η-almost all (Yz, νz)are standard. LetΓ y^a (Y, ν)be given by letting Γ act likeaz

onYz. We write a = ∫

Z azdη(z). We then have a map φ :Y → Z given by letting φ(y)be the uniquezsuch thaty ∈Yz. This is clearly a factor map fromatoιZ,ηand ν = ∫

Zνzdη(z)is the disintegration of ν overηvia φ. Thus Theorem 3.12 in [74]

guarantees that ifbz are actions ofΓon(Yz, νz)withbz ∼ az then if b=∫

Z bzdη(z) we havea ∼ b. Therefore this construction gives a well-defined weak equivalence class of actions of γ. If we restrict (Yz, νz) to be standard, then we in fact have a mapping from the space M(A∼(Γ,X, µ))of probability measures on A∼(Γ,X, µ)to A∼(Γ,X, µ).

Lemma 2.4.2. For anyn,k, and(Z, η)and measurable assignmentz7→ az, we have Cn,k ∫

Zazdη(z)

⊆ cch Ð

z∈ZCn,k(az) . Proof. Fixn,kand leta= ∫

Zazdη(z). Let(Xz, µz)by the underlying measure space ofaz. LetL be a countable dense subset of MALG

z∈Z Xz,∫

Z µzdη(z)

, so that

L^k is dense in the space ofk-partitions ofÃ

z∈Z Xz. Then{MA(a)}_A∈Lnis dense in Cn,k ∫

Zazdη(z)

, so it suffices to show that each MA(a) ∈ cch Ð

z∈ZC(az) . For eachA, the function fA : Z → R^n×k×k given byz 7→ MA_z(az)is a Borel function, whereA_zis the partition ofXzgiven by(A∩Xz)_A∈A. ThusMA(a)= ∫

Z fA(a)dη(z).

We may assume thatZ carries a Polish topology such that fA is continuous for all A ∈ Lⁿ. Choose a sequence of measures(νi)^∞

i=1such thatνihas finite support and νi → ηin the topology of M(Z), the space of all Borel probability measures onZ. If we writeνi =Í^j(i)

j=1αjδz_j then

∫

fA(z)dνi(z)=

j(i)

j=1

αjfA(zj) ∈ch Ø

z∈A

C(az)

! .

Sinceνi → η, we have

∫

fA(z)dνi(z) →

∫

fA(z)dη(z)

which proves the lemma.

2.5 The structure of the space of weak equivalence classes for amenable acting

Dalam dokumen Approximation and classiﬁcation in the ergodic theory of nonamenable groups (Halaman 38-43)