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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. To see ha he de ini ion makes sense,
ON ONE-DIMENSIONALITY OF METRIC MEASURE SPACES 15
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