On one-dimensionality of metric measure spaces

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY-NC-ND 4.0 h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ On one-dimensionali y o me ic measu e spaces © 2020 Ame ican Ma hema ical Socie y Accep ed e sion (Final d a ) Schul z, Timo Schul z, T. (2021). On one-dimensionali y o me ic measu e spaces. P oceedings o he Ame ican Ma hema ical Socie y, 149(1), 383-396. h ps://doi.o g/10.1090/p oc/15162 2021 ON ONE-DIMENSIONALITY OF METRIC MEASURE SPACES TIMO SCHULTZ Abs ac . In his pape , we p o e ha a me ic measu e space which has a leas one open se isome ic o an in e al, and o which he (possibly non-unique) op imal ans- po map exis s om any absolu ely con inuous measu e o an a bi a y measu e, is a one-dimensional mani old (possibly wi h bounda y). As an immedia e co olla y we ob- ain ha i a me ic measu e space is a e y s ic CD(K, N) -space o an essen ially non-b anching MCP (K, N)-space wi h some open se isome ic o an in e al, hen i is a one-dimensional mani old. We also ob ain he same conclusion o a me ic measu e space which has a poin in which he G omo -Hausdo angen is unique and isome ic o he eal line, and o which he op imal anspo maps no only exis bu a e unique. Again, we ob ain an analogous co olla y in he se ing o essen ially non-b anching MCP (K, N)- spaces. 1. In oduc ion The s ong in e play be ween op imal mass anspo a ion and (me ic) geome y has been acknowledged in he las ew decades leading o a g ea numbe o applica ions o example in he s udy o geome ic and analy ic inequali ies, in desc ibing and de ining cu a u e bounds, and in he egula i y heo y o pa ial di e en ial equa ions. The op imal anspo heo y is use ul bo h in gene alising classical esul s om he heo y o smoo h mani olds o possibly singula spaces, and in ob aining new esul s e en in he smoo h se ing. In he p esen pape we will use ools om op imal anspo heo y o ob ain global opological/geome ic in o ma ion abou a me ic (measu e) space om in o ma ion nea a single poin in he space. Mo e p ecisely, we will use he exis ence – and in some cases uniqueness – o an op imal anspo map oge he wi h one-dimensionali y a (Theo em 3.10 and Co olla y 3.11), o nea (Theo em 3.1, Co olla y 3.2 and Theo em 3.5), a poin in he space o p o e ha he space in ques ion is a one-dimensional mani old. He e, by one-dimensionali y a a poin , we mean ha he G omo -Hausdo angen a ha poin is unique and isome ic o he eal line, and by one-dimensionali y nea a poin , we mean ha he poin has an open neighbou hood isome ic o an open in e al. Such a esul was i s p o en in he se ing o Ricci limi spaces in [9] by Honda and gene alised o he se ing o RCD∗(K, N)-spaces in [12] by Ki abeppu and Lakzian. In bo h pape s, i is p o en ha he unde lying space sa is ying he syn he ic Ricci cu a u e lowe Da e: Ap il 27, 2020. 2000 Ma hema ics Subjec Classi ica ion. P ima y 53C23. Key wo ds and ph ases. Op imal anspo , Ricci cu a u e, me ic measu e spaces, G omo –Hausdo angen s. 1 a Xi :1912.01579 2 [ma h.MG] 24 Ap 2020 2 TIMO SCHULTZ bound in ques ion, is a one-dimensional mani old i i is one-dimensional a a single poin . These wo pape s sha e a common (and na u al) iewpoin coming om he s uc u e heo y: in bo h se ings one could w i e he me ic measu e space up o a ze o measu e se as a union o se s Rkin which one has he exis ence and uniqueness o angen s isomo phic o k-dimensional Euclidean space [5, 16]. In ou se ing such a decomposi ion o he space canno be ue in gene al, no leas because o he allowance o Finsle ian ( ype) s uc u es. No e ha i is s ill meaning ul o ask wha can be concluded om he exis ence o a one- dimensional pa e en i we would impose Finsle ian ype beha iou , since a p io i he dimension o he space needs no be cons an . The s udy o he p esen pape was pa ially mo i a ed by he esul s conce ning he ex- is ence o op imal anspo maps on he spaces ha ing syn he ic Ricci cu a u e bounded om below. A no ion o Ricci cu a u e lowe bound o (possibly singula ) me ic measu e spaces – he so-called CD(K, N)-condi ion (cu a u e dimension condi ion) – was in o- duced in he seminal wo ks o S u m [21, 22] and o Lo and Villani [14]. The exis ence and uniqueness o he op imal map in he se ing o spaces wi h syn he ic Ricci cu a u e lowe bounds was i s p o en by Gigli in [7] o non-b anching CD(K, N)-spaces, and hen gene alised o s ong CD(K, N)-spaces by Rajala and S u m in [18] and by Gigli, Rajala and S u m in [8]. In hei pape , Rajala and S u m in oduced he no ion called essen ial non-b anchingness, which u ned ou o be a use ul gene alisa ion o he non-b anching assump ion on me ic measu e spaces. In [4], Ca alle i and Mondino p o ed he exis ence and uniqueness o op imal anspo maps in MCP(K, N)-spaces i one assumes ha he unde lying me ic measu e space is essen ially non-b anching. Then in [10], Kell gene - alised he esul o spaces sa is ying e en weake e sion o cu a u e lowe bound, namely o he se ing o quali a i ely non-degene a e spaces (s udied by Ca alle i and Huesmann in he non-b anching case in [2]) – s ill unde he essen ial non-b anching assump ion. Heu is ically, he non-b anching assump ion p e en s he geodesics o an op imal plan o in e sec a in e media e imes, while he cu a u e lowe bound assump ion o ces hem o in e sec when he plan is assumed no o be induced by a map, hence he exis ence and uniqueness o op imal maps is ob ained by combining hese wo. The e o e, while he uniqueness o he op imal map is los i he e exis s an essen ial amoun o b anching geodesics, one migh s ill pu sue he exis ence o such a map. This app oach was aken in [19] (and con inued in [20]), whe e he au ho p o ed he exis ence o op imal anspo maps in he se ing o so-called e y s ic CD(K, N)-spaces. We ema k ha while in gene al (b anching) MCP(K, N)-space he exis ence o an op imal anspo map migh ail by he example in [11], i is s ill no known whe he op imal maps exis in gene al CD(K, N)-spaces. Acknowledgemen s. The au ho acknowledges he suppo by he Academy o Finland, p ojec #314789, and hanks he anonymous e e ee o ca e ully eading he pape . 2. P elimina ies Fo he pu poses o his pape , we will always assume ha (X, d, m) is a me ic measu e space which is a comple e, locally compac and sepa able leng h space (X, d) equipped wi h ON ONE-DIMENSIONALITY OF METRIC MEASURE SPACES 3 a locally ini e measu e m. We will also assume ha sp m=X. The space o (cons an speed, leng h minimising) geodesics pa ame ised by [0,1] is deno ed by Geo(X), and i is equipped wi h he sup emum dis ance. 2.1. Op imal mass anspo a ion. In his sec ion, we in oduce he basic no ions o op imal anspo heo y which se he basis o he pape . In addi ion, we es ablish in P oposi ion 2.1 a sub le de ail abou he exis ence and uniqueness o op imal anspo maps in he case o non-geodesic spaces. In he main esul s o he pape we a e assuming he exis ence o an op imal anspo map o he Monge–Kan o o ich p oblem wi h quad a ic cos . The eason o such a choice o he cos unc ion lies in he connec ion be ween Wasse s ein geodesics and op imal dynamical anspo plans. No e ha one could ob ain simila esul s by conside ing cos unc ions o he o m dp o p∈(1,∞) di e en om 2, since he ep esen a ion o Wasse s ein geodesics by measu es on he space o geodesics, and he co esponding exis ence esul s o anspo maps emain ue in his case. The quad a ic Monge–Kan o o ich p oblem eads as ollows. Le µ0, µ1∈ P(X) be Bo el p obabili y measu es on X. Conside he minimisa ion p oblem W2 2(µ0, µ1):= in Zd2(x, y) dσ(x, y), whe e he in imum is aken o e all anspo plans σ, ha is, o e all Bo el p obabili y measu es σ∈ P(X×X) wi h µ0and µ1as ma ginals (P1 #σ=µ0,P2 #σ=µ1). I is a s anda d ac in op imal anspo heo y ha in he se ing o comple e and sepa able me ic spaces, he abo e in imum is in ac a minimum. We will call a minimise o he p oblem an op imal ( anspo ) plan, and deno e he se o all op imal plans by Op (µ0, µ1). The op imali y o a plan can be cha ac e ised by he so-called c-cyclical mono onici y in he ollowing way. Le σbe a anspo plan be ween measu es µ0and µ1 o which W2 2(µ0, µ1)<∞. Then σis op imal i and only i i is concen a ed on a c-cyclically mono one se , ha is, i he e exis s a se Γ ⊂X×Xso ha σ(Γ) = 1, and X i d2(xi, yi)≤X i d2(xi, yτ(i)) o any ini e se {(xi, yi)}i⊂Γ and o any pe mu a ion τ. The unc ion W2(·,·) de ines a me ic on he subse P2(X)⊂ P(X) o p obabili y mea- su es wi h ini e second momen , ha is µ∈ P(X) wi h Rd2(x, x0) dµ(x)<∞ o some x0∈X. The dis ance W2is he so-called Wasse s ein dis ance (o mo e p ecisely he 2-Wasse s ein dis ance). Since Xis a comple e, sepa able and – by Hop –Rinow heo em – geodesic space, so is he Wasse s ein space (P2(X), W2). Mo eo e , a cu e (µ )⊂ P2(X) is a geodesic i , and only i , he e exis s a measu e π∈ P(Geo(X)) so ha µ = (e )#πand (e0, e1)#π∈Op (µ0, µ1) [13]. He e e : Geo(X)→Ris he e alua ion map γ7→ γ . Such a πis called an op imal (geodesic) plan, and he se o all op imal geodesic plans is deno ed by Op Geo(µ0, µ1). 4 TIMO SCHULTZ We say ha he op imal plan σ∈Op (µ0, µ1) is induced by a map i he e exis s a Bo el map T:X→X×Xsuch ha σ=T#µ0and P1◦T= id. Analogously, π∈Op Geo(µ0, µ1) is induced by a map T:X→Geo(X), i π=T#µ0and e0◦T= id. In he se ing o geodesic spaces, one can always li he op imal plan σ∈Op (µ0, µ1) o an op imal geodesic plan π∈Op Geo(µ0, µ1) by making a measu able selec ion o geodesics o each pai o poin s in X. The e o e, he ques ion o exis ence o op imal maps in he le el o plans in Op (µ0, µ1) and o geodesic plans in Op Geo(µ0, µ1) a e equi alen . Fu he mo e, i he op imal plan π∈Op Geo(µ0, µ1) is unique, so is σ∈Op (µ0, µ1). Since he amewo k o his pape is ha o comple e and locally compac me ic spaces, being a leng h space is equi alen o being geodesic. Howe e , some imes i is mo e na u al o d op he assump ion o local compac ness and s ill impose condi ions implying he leng h s uc u e o he space ( his is he case o example in he CD(K, ∞)-se ing). We ema k he ollowing connec ion be ween exis ence o op imal maps o anspo plans and o geodesic anspo plans in he abo e-men ioned non-geodesic case, which may be o independen in e es . P oposi ion 2.1. Le (X, d, m)be a me ic measu e space (possibly non-geodesic and non- locally-compac ). Assume ha o all µ0, µ1∈ Pac 2(X) he se Op Geo(µ0, µ1)is non-emp y, and ha each op imal dynamical plan π∈Op Geo(µ0, µ1)is induced by a map. Then, o any µ0, µ1∈ Pac 2(X), e e y op imal plan σ∈Op (µ0, µ1)is induced by a map, g an ing also he uniqueness o he op imal plan. In pa icula , e e y op imal plan σ∈Op (µ0, µ1),µ0, µ1∈ Pac 2(X), can be li ed o a unique dynamical plan π∈Op Geo(µ0, µ1) o which σ= (e0, e1)#π. P oo . Le σ∈Op (µ0, µ1)⊂ P(X2). Suppose ha σis no induced by a map. Then he e exis s a µ0-posi i e measu e Bo el se A⊂Xsuch ha σxis no a Di ac mass o any x∈A, whe e {σx}x∈Xis a disin eg a ion o σwi h espec o he p ojec ion P1. W i e A=[ i,j∈N Aij, whe e Aij :={x∈A:σx(B(ξi,1/j)) ∈(0,1)}, and {ξi}i∈Nis dense in X. Se s Aij a e measu able, since he maps x7→ σx(B(ξi,1/j)) a e measu able o all i, j ∈Nby he disin eg a ion heo em. Since µ0(A)>0, he e exis i0and j0, such ha µ0(Ai0j0)>0. De ine now σ1and σ2as σ1:=σ|X×B(ξi0,1 j0), σ2:=σ|X×(X B(ξi0,1 j0)). Fo k∈ {1,2}, de ine ˆσkas he measu e o which Z dˆσk:=Z min{ρ1 0, ρ2 0} ρk 0 ◦P1dσk, o all posi i e Bo el unc ions , whe e ρk 0is he densi y o P1 #σkwi h espec o he e e ence measu e m. He e we use con en ion min{ρ1 0,ρ2 0} ρk 0 = 0, when ρk 0= 0. Since he unc ion min{ρ1 0,ρ2 0} ρk 0 ∈[0,1], we ha e ha ˆσkis a well-de ined and ini e measu e. By he ON ONE-DIMENSIONALITY OF METRIC MEASURE SPACES 5 de ini ion o Ai0j0, we ha e ha ρk 06= 0 o µ0-almos e e y x∈Ai0j0. In pa icula , by he absolu e con inui y o µ0, he e exis s an m-posi i e measu e se whe e ρk 06= 0, and hus ˆσkis non- i ial o k∈ {1,2}. Fo k∈ {1,2}and j∈ {0,1}, w i e ˆµk j:= Pj+1 #ˆσk. Then we ha e ha ˆµ1 0= ˆµ2 0, ˆµ1 1⊥ˆµ2 1, and ˆσk∈Op (ˆµk 0,ˆµk 1) (o mo e p ecisely, 1 Nˆσk∈Op ( 1 Nˆµk 0,1 Nˆµk 1), whe e Nis he no malisa ion cons an N= ˆµ1 0(X) = ˆµ2 0(X)). Since ˆσkσ, we ha e ha ˆµk jm. Thus by assump ion, o k∈ {1,2}, he e exis s an op imal dynamical plan πk∈Op Geo(ˆµk 0,ˆµk 1). On he o he hand, since (ˆσ1+ ˆσ2)σand since Zd2(x, y) d(ˆσ1+ ˆσ2)(x, y) = Zd2(x, y) d((e0+e1)#(π1+π2))(x, y), we ha e ha π1+π2is an op imal dynamical plan be ween absolu ely con inuous measu es 2ˆµ1 0and ˆµ1 1+ ˆµ2 1 ha is no induced by a map. Hence, we a i e o a con adic ion wi h he assump ion.  Rema k 2.2.In he abo e p oo he ull exis ence and uniqueness o op imal geodesic plans is no needed, bu ins ead exis ence and uniqueness o op imal geodesic plans inside some linea ly con ex subse o P(Geo(X)). In pa icula , he p oo can be adap ed o p o e he uniqueness o he op imal plan in Op (µ0, µ1) be ween absolu ely con inuous measu es in he essen ially non-b anching CD(K, ∞)-spaces by using he esul [10, Co olla y 5.22] o he exis ence and uniqueness o op imal geodesic plans among all plans π o which µ = (e )#πis absolu ely con inuous o all ∈[0,1]. 2.2. (Measu ed) G omo –Hausdo angen s. The e a e di e en no ions o blow- ups o me ic (measu e) spaces. The one ha we will use is based on a con e gence o poin ed me ic spaces in he G omo –Hausdo sense: De ini ion 2.3 (poin ed G omo –Hausdo con e gence).Le (Xi)i∈N= (Xi, di, xi)i∈Nbe a sequence o poin ed, comple e, sepa able and locally compac geodesic spaces. Then Xi−→ X∞= (X∞, d∞, x∞), i o all R > 0 he e exis s a sequence εi→0 and (1, εi)- quasi-isome ies i:¯ B(xi, R)→¯ B(x∞, R) wi h xi7→ x∞. He e being (1, ε)-quasi-isome y is de ined by equi ing ha is 1-biLipschi z up o an addi i e cons an ε > 0 wi h an ε-dense image. Tangen s o a me ic space Xa a poin x∈Xa e hen ob ained by looking a sequences o he o m (X, λid, x) wi h λi→ ∞. De ini ion 2.4. Le Xbe a comple e, sepa able and locally compac geodesic space, and le x∈X. A poin ed me ic space (Y, dY, y) is a G omo –Hausdo angen o Xa x, (Y, dY, y)∈Tan(X, x), i he e exis s a sequence λi→ ∞ so ha (X, λid, x)−→ (Y, dY, y) in he poin ed G omo –Hausdo opology. Fo doubling me ic spaces he se o angen s a each poin is non-emp y. On he o he hand, in gene al angen s a e no unique. We poin ou ha he e exis no ions o angen s o me ic measu e spaces ha ake in o accoun he con e gence o he (no malised) measu e, which a e in many cases mo e 6 TIMO SCHULTZ sui able o he s udy o me ic measu e spaces. Howe e , o ou pu poses i is enough o conside he con e gence as a me ic concep (keeping in mind ha we a e assuming sp m=X). 3. One-dimensionali y o me ic measu e spaces In his sec ion we p o ide a gene alisa ion o he ollowing heo em Theo em ([12, Theo em 3.7]).Le (X, d, m)be an RCD∗(K, N)space o K∈Rand N∈(1,∞). Assume ha he e exis s a poin x0∈Xsuch ha he e exis s a unique (up o an isomo phism) measu ed G omo -Hausdo angen o (X, d, m)a x0isomo phic (as a poin ed me ic measu e space) o (R,|·|, cL1,0). Then o any x∈X, he e exis s a posi i e numbe ε > 0such ha B(x, ε)is isome ic o (−ε, ε)o o [0, ε). In Theo em 3.10 we s a e he esul implying one-dimensional mani old s uc u e om assump ions on he one-dimensionali y a a poin a in ini esimal le el, analogously o he o iginal esul , when imposing exis ence and uniqueness o op imal anspo maps. In Theo em 3.1 we gi e a local coun e pa o he esul o he case whe e uniqueness o anspo maps is los . P oo s p esen ed he e ake ad an age o he exis ence o op imal anspo maps (assump ion which may be jus i ied by he esul s in [8, 18, 4, 10, 19]), and by ha simpli y he ones gi en o Theo em 3.7 (and hence o Theo em 1.1) in [12]. Theo em 3.1. Le (X, d, m)be a me ic measu e space wi h he ollowing p ope ies: (1) Fo e e y µ0∈ Pac 2(X)and µ1∈ P2(X), he e exis s π∈Op Geo(µ0, µ1) ha is induced by a map om µ0. (2) The e exis s a poin x∈X, and a neighbou hood B(x, )isome ic o an open in e al in R. Then Xis a one-dimensional mani old, possibly wi h bounda y. Due o he exis ence o op imal anspo maps in e y s ic CD(K, N) -spaces [20], and in essen ially non-b anching MCP(K, N)-spaces [4], we ge he ollowing immedia e co olla y o Theo em 3.1. Co olla y 3.2. Le (X, d, m)be a e y s ic CD(K, N)-space (N∈(1,∞)), o essen ially non-b anching MCP(K, N)-space (o ess. nb., quali a i ely non-degene a e space, see [10]). Suppose ha he e exis s a poin x∈X, and a neighbou hood B(x, )isome ic o an open in e al in R. Then Xis a one-dimensional mani old, possibly wi h bounda y. I is wo h no icing, ha while Co olla y 3.2 is known o be alse in gene al MCP(K, N)- space by he example gi en by Ke e e and Rajala in [11], i emains s ill open in gene al CD(K, N)-space: Ques ion 1. Le (X, d, m) be a CD(K, N)-space, and B(x, )⊂Xisome ic o an open in e al. Is Xa mani old (possibly wi h bounda y)? P oo o Theo em 3.1. The beginning o he p oo goes jus as ha o Theo em 3.7 in [12]. Deno e by F he se o all he poin s ha ha e a neighbou hood isome ic o an open ON ONE-DIMENSIONALITY OF METRIC MEASURE SPACES 7 in e al. Clea ly, Fis an open se in X. Suppose now ha X F 6=∅. Since Fis assumed o be non-emp y, we deduce ha Fis no closed, in pa icula i is no a ci cle. Le γ: (−a, b)→X,a, b ∈(0,∞], be a locally minimising, uni speed cu e o which γ0=x,xbeing as in he assump ion (2), and γ(−a, b) is he maximal connec ed subse o Fcon aining x, and le ε > 0 be such ha γ|(−ε,ε)is an isome y. I a=b=∞, we ha e ha γ(−a, b) = Xby he ollowing a gumen . Suppose ha he e exis s a limi poin y∈Im γ Im γo γ. Le ( i)⊂Rbe a sequence so ha γ i−→ y. We may assume ha iis inc easing. I ( i) is bounded, hen he e exis s ∞so ha i−→ ∞, and hence by con inui y we would ha e ha y=γ ∞. Hence i−→ ∞. Le now αbe a geodesic om x o y, and le s:= in { :α /∈Im γ}. We know ha s > 0, since he neighbou hood o xis isome ic o an open in e al. On he o he hand, since αis a geodesic and l(γ) = ∞, we know ha he e exis s a sequence (si) so ha si−→ s∞<∞, and γsi−→ αs. Hence, γs∞=αs. This is in con adic ion wi h he de ini ion o s, since γs∞has a neighbou hood o he o m γ(s∞−δ, s∞+δ). Thus, γ(−a, b) is open and closed, and hence γ(−a, b) = Xgi ing a con adic ion. By he abo e, we may assume ha b < ∞. Le y∈X Im γ,αbe a uni speed geodesic om x o y, and, as abo e, s:= in { :α /∈Im γ}. Since by he a gumen s abo e, γ 6=αs o any ∈(−a, b), he e exis s a sequence ( i) ha (we may assume wi hou loss o gene ali y o) con e ge o bso ha γ i−→ αs. By he maximali y o Im γ, we know ha B(αs, ) (Im γ∪Im α)6=∅ o any > 0. Le now z∈B(αs, ) (Im γ∪Im α), whe e l(α)> > 0 is chosen small enough so ha any geodesic om x o zgoes h ough he poin αs. We a e eady o a i e o a con adic ion wi h (1). Le βbe a uni speed geodesic om x o z. Take 1> s so ha d(αs, α 1) = d(αs, z), and de ine measu es µ0:=1 m(γ([0,ε))) m|γ([0,ε))∈ Pac 2(X), and µ1:=1 2((α◦ es 1 0)#L1+ (β◦ es 1 0)#L1)∈ P2(X), whe e 0is chosen so ha α 6=β o e e y ∈[ 0, 1]. Then, by using c-cyclical mono onici y as ollows, we deduce ha any plan om µ0 o µ1canno be gi en by a map, gi ing a con adic ion wi h he assump ion (1). Indeed, i σ∈Op (µ0, µ1) is induced by a map, he e exis s a c-cyclically mono one se Γ ⊂X×X so ha σ(Γ) = 1, and Γ˜x:={y∈X: (˜x, y)∈Γ}is a single on o all ˜x∈P1(Γ). By he de ini ion o µ1, and he ac ha σ(Γ) = 1, we ha e ha he e exis (x1, α( )),(x2, β( )) ∈Γ wi h some ∈[ 0, 1]. Since Γ˜xis a single on o all ˜x, we deduce ha x16=x2. Then by he de ini ions o µ0and µ1, he e exis s ˜xbe ween x1and x2so ha (˜x, y)∈Γ o some y∈X, and so ha β( )6=y6=α( ). Suppose ha y=α(˜ ), wi h ˜ < ( he o he cases a e analogous). Then d2(x1, α( )) + d2(˜x, y)=[d(˜x, α( )) + d(˜x, x1)]2+ [d(x1, y)−d(˜x, x1)]2 > d2(˜x, α( )) + d2(x1, y), which con adic s he c-cyclical mono onici y o he se Γ.  Rema k 3.3.The assump ion ha he e exis s an op imal map o all µ1, ins ead o only he absolu ely con inuous ones, is c ucial in Theo em 3.1. This can be seen by aking h ee non-a omic, mu ually singula p obabili y measu es wi h ull suppo s on he in e al [0,1], and pushing hem o di e en b anches o a ipod (see [10, Example in Sec ion 3] o 8 TIMO SCHULTZ mo e de ails). The me ic measu e space ob ained in his way sa is ies he assump ion o exis ence o anspo maps be ween absolu ely con inuous measu es, bu does no sa is y he conclusion o Theo em 3.1. One migh wonde , whe he a e elaxing he assump ion (1) o conce n only absolu ely con inuous measu es µ1one could s ill conclude ha he space is one-dimensional in some app op ia e sense (as in he case o he abo e-men ioned ipod). The ollowing example shows ha his is no he case in gene al. Example 3.4. We will cons uc a me ic measu e space ha ing – as a me ic space – a one-dimensional pa (a line segmen ), and a wo-dimensional pa (a ci cula sec o ) which has he p ope y ha he op imal plan be ween any wo absolu ely con inuous Bo el p obabili y measu es is unique and induced by a map (see Figu e 1). Le us i s de ine µs µ Figu e 1. The space sa is ying he weakened assump ions o Theo em 3.1, bu ailing he conclusion. auxilia y measu es on he uni squa e I×I. Le : 2N→Ibe a map de ined as (xi)i∈N7→ lim N→∞ 1 N N X i=1 xi. De ine a amily o measu es {µ } ∈Ion I×Ias a push o wa d o he measu es ν :=⊗ N [(1 − )δ0+ δ1] unde he g aph map (ι, ), whe e ι: 2N→Iis he Bo el isomo phism (up o emo ing a coun able se ) (xi)i∈N7→ Pixi2−i. Finally, de ine a measu e ˜ mon I×Iby se ing Zgd˜ m:=ZIZI×I gdµ dL1( ) = ZIZI×{ } gdµ dL1( ) o e e y posi i e Bo el unc ion g. He e he second equali y is due o he ac ha ν ( −1( )) = 1 by he s ong law o la ge numbe s. 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