Observable concentration of mm-spaces into nonpositively curved manifolds

0.43 MB·English

by  Kei Funano

8 OBSERVABLE CONCENTRATION OF MM-SPACES INTO
0 NONPOSITIVELY CURVED MANIFOLDS
0
2
n KEI FUNANO
a
J

0
3
Abstract. Themeasureconcentrationpropertyofanmm-spaceX isroughlydescribed
] asthatany1-LipschitzmaponX toametricspaceY isalmostclosetoaconstantmap.
G The target space Y is called the screen. The case of Y = R is widely studied in many
M literature (see [11], [14], [20], [21], [25], [28], [29] and their references). M. Gromov
developed the theory of measure concentration in the case where the screen Y is not
.
h necessarily R (cf. [9], [10], [11]). In this paper, we consider the case where the screen Y
t
a is a nonpositivelycurvedmanifold. We also show that if the screenY is so big, then the
m mm-space X does not concentrate.
[

2 1. Introduction
1
v Let µ be the volume measure on the n-dimensional unit sphere Sn in Rn+1 normalized
n
5
as µ (Sn) = 1. In 1919, P. L´evy proved that for any 1-Lipschitz function f : Sn R and
3 n
→
5 any ε > 0, the inequality
1
0 (1.1) µn x Sn f(x) mf ε 2 e−(n−1)ε2/2
7 { ∈ | | − | ≥ } ≤
0 holds, where mf is som(cid:0)e constant determined by f(cid:1). For any fixed ε > 0 the right-hand
/
h side of the above inequality converges to zero as n . This means that any 1-
→ ∞
at Lipschitz function on Sn is almost close to a constant function for sufficiently large n N.
∈
m This high dimensional concentration phenomenon of functions was first extensively used
: and emphasized by V. D. Milman in his investigation of asymptotic geometric analysis.
v
i He used L´evy’s inequality (1.1) for a short proof of Dvoretzky’s theorem on Euclidean
X
section of convex bodies (cf. [18]). The same year later he extended L´evy’s result to some
r
a nonspherical spaces and then pushed forward the idea of concentration of functions as a
general unifying principle (cf. [17], [19]). Nowadays, the concentration theory of functions
is widely studied in many literature and blend with various areas of mathematics (see
[11], [14], [20], [21], [25], [28], [29] and references therein for further information).
In 1999, M. Gromov established a theory of concentration of maps into general met-
ric spaces by introducing the notion of the observable diameter in [11]. He settled the
following definition.
Date: February 2, 2008.
2000 Mathematics Subject Classification. 31C15,53C21, 53C23.
Key words and phrases. concentration of maps, nonpositively curved manifold, observable diameter.
This work was partially supported by ResearchFellowships of the JapanSociety for the Promotionof
Science for Young Scientists.
1
2 KEI FUNANO
Definition 1.1. Let Y be a metric space and ν a Borel measure on Y such that m :=
Y
ν (Y) < + . We define for any κ > 0
Y
∞
diam(ν ,m κ) := inf diamY Y Y is a Borel subset such that ν (Y ) m κ
Y 0 0 Y 0
− { | ⊆ ≥ − }
and call it the partial diameter of ν .
Y
An mm-space is a triple (X,d,µ), where d is a complete separable metric on a set X
and µ a finite Borel measure on (X,d).
Definition 1.2 (Observable diameter). Let (X,d,µ) be an mm-space and Y a metric
space. For any κ > 0 we define the observable diameter of X by
Lip
diam(X 1 Y,m κ) := sup diam(f (µ),m κ) f : X Y is a 1-Lipschitz map ,
−→ − { ∗ − | → }
where f (µ) stands for the push-forward measure of µ by f. The target metric space Y
∗
is called the screen.
The idea of the observable diameter comes from the quantum and statistical mechan-
ics, that is, we think of µ as a state on a configuration space X and f is interpreted
as an observable. We define a sequence X of mm-spaces is a L´evy family if
{ n}∞n=1
Lip
diam(X 1 R,m κ) 0 as n for any κ > 0, where m is the total mea-
n n n
−→ − → → ∞
sure of the mm-space X . This is equivalent to that for any ε > 0 and any sequence
n
f : X R of 1-Lipschitz functions, we have
{ n n → }∞n=1
µ ( x X f (x) m ε ) 0 as n ,
n { ∈ n | | n − fn| ≥ } → → ∞
wherem isasomeconstantdeterminedbyf . TheL´evyfamiliesarefirstintroducedand
fn n
analyzed from a geometric point of view by Gromov and Milman in [12]. The inequality
(1.1) shows that the sequence Sn is a L´evy family. Gromov proved in [11] that
{ }∞n=1
Lip
diam(X 1 Y,m κ) 0 as n for any κ > 0 and for a L´evy family X if
n−→ n − → → ∞ { n}∞n=1
the screen Y is a compact metric space or a Euclidean space. He also discussed the case
where the screens are Euclidean spaces whose dimensions go to infinity by considering the
barycenters of the push-forward measures (see Theorem 4.31). His barycenter method
also goes well if the screens are nonpositively curved manifolds (see Section 4). In [10], he
considers and analyzes the questions of isoperimetry of waists and concentration for maps
from a unit sphere to a Euclidean space (see Theorem 4.27). In the recent work [15], M.
Lip
Ledoux and K. Oleszkiewicz estimated the observable diameter diam(X 1 Rk,m κ)
−→ −
provided that the mm-space X has a Gaussian concentration (see Theorem 4.22). In our
previous paper [8], the author studied the case where the screen Y is a metric space with
a doubling measure.
Inthispaper, inspired by Gromov’sstudy, westudy concentration phenomenon of maps
into nonpositively curved manifolds. In particular, we consider the case where dimensions
of screens go to infinity. We denote by n the set of all n-dimensional complete simply
NM
connected Riemannian manifolds with nonpositive sectional curvature. One of our main
theorems is the following:
3
Theorem 1.3. Let X be a sequence of mm-spaces. We assume that a sequence
{ n}∞n=1
a(n) of natural numbers satisfies that for any κ > 0
{ }∞n=1
(1.2) diam(X Lip1 Ra(n),m κ) 0 as n .
n n
−→ − → → ∞
Then for any κ > 0, we have
sup diam(X Lip1 N,m κ) N a(n) 0 as n .
n n
{ −→ − | ∈ NM } → → ∞
IntheproofofTheorem1.3,wefindapointinascreenN whichisakindofbarycenterof
the push-forward measure on N, and prove that the measure concentrates to the point by
the delicate discussions comparing N with both an Euclidean space and a real hyperbolic
space.
In [9], Gromov introduced the notion of Lp-concentration of maps from mm-spaces into
metric spaces. We recall its definition. Let (X,d ,µ ) be an mm-space and (Y,d ) a
X X Y
metric space. Given a Borel measurable map f : X Y and p (0,+ ), we put
→ ∈ ∞
1/p
p
Vp(f) := dY f(x),f(x′) dµX(x) dµX(x′) ,
(cid:16)Z ZX×X (cid:17)
(cid:0) (cid:1)
V (f) := inf r > 0 (µX µX)( (x,x′) X X dY(f(x),f(x′)) r ) = 0 .
∞ | × { ∈ × | ≥ }
Let {Xn}∞n=1 be(cid:8)a sequence of mm-spaces and {Yn}∞n=1 a sequence of metric space(cid:9)s. For
any p (0,+ ], we say that a sequence f : X Y of Borel measurable maps
∈ ∞ { n n → n}∞n=1
Lp-concentrates if V (f ) 0 as n .
p n
→ → ∞
We easily see that Lp-concentration of maps implies the concentration of maps (see
Lemma 2.19 and Corollary 2.20). In [9], Gromov got several L2-concentration inequalities
of maps (see for example, Theorem 4.28).
Given an mm-space X and a metric space Y we define
ObsLp-Var (X) := sup V (f) f : X Y is a 1-Lipschitz map ,
Y p
{ | → }
and call it the observable Lp-variation of X. We get the following Lp-concentration result:
Theorem 1.4. Let X be an mm-space and p 1. Then, we have
≥
(1.3) ObsLp-VarN(X) 2ObsLp-VarRn(X)
≤
for any n N and N n. In particular, if a sequence X of mm-spaces and a
∈ ∈ NM { n}∞n=1
sequence a(n) of natural numbers satisfy that
{ }∞n=1
(1.4) ObsLp-Var (X ) 0 as n
Ra(n) n
→ → ∞
for some p 1, then we have
≥
(1.5) sup ObsLp-Var (X ) N a(n) 0 as n .
N n
{ | ∈ NM } → → ∞
In [9, Section 13], Gromov got ObsL2-VarN(X) √2ObsL2-VarRn(X). Note that this
≤
Gromov’s inequality implies better estimate than our inequality (1.3) (see also Remark
3.7). Our proof is an analogue to his proof.
The following theorem includes the case of 0 < p < 1:
4 KEI FUNANO
Theorem 1.5. Let X be a sequence of mm-spaces with finite diameter. Assume
{ n}∞n=1
that a sequence a(n) of natural numbers satisfies that supm < + and
{ }∞n=1 n N n ∞
∈
κ
(1.6) diam X Lip1 Ra(n),m 0 as n
n−→ n − (diamX )p → → ∞
n
(cid:16) (cid:17)
for any κ > 0 and some p > 0. Then we have
(1.7) sup ObsLp-Var (X ) N a(n) 0 as n .
N n
{ | ∈ NM } → → ∞
Note that the assumption (1.6) directly implies (1.2). Actually, (1.6) leads to (1.4) (see
Corollary 2.22). We do not know whether the assumption (1.4) for 0 < p < 1 implies (1.7)
or not. By using Gromov’s observation in [11], we prove a similar result of Theorems 1.4
and 1.5 (see Proposition 4.5 and Remark 4.6).
The next proposition says that if the screen Y is so big that the mm-space X which
satisfies some homogeneity condition, can isometrically be embedded into Y, then its
Lip
observable diameter diam(X 1 Y,m κ) is not close to zero.
−→ −
Proposition 1.6. Let (X ,d ,µ ) be a L´evy family with inf m > 0 and Y
{ n n n }∞n=1 n N n { n}∞n=1
∈
a sequence of metric spaces satisfying the following conditions (1) (3).
−
(1) X = Suppµ is connected.
n n
(2) For any r > 0 and n N, all measures of closed balls in X with radius r are
n
∈
mutually equal.
(3) For each n N, there exists an isometric embedding from X to Y .
n n
∈
Then, for any κ with 0 < κ < inf m , we have
n
n N
∈
1
Lip
liminfdiam(X 1 Y ,m κ) liminfdiamX .
n n n n
n −→ − ≥ 2 n
→∞ →∞
From Proposition 1.6, we obtain the following corollary:
Corollary 1.7. For any κ with 0 < κ < 1, we have
liminfdiam(Sn Lip1 Rn+1,1 κ) > 0.
n −→ −
→∞
Note that the dimension n + 1 of the screen can be replaced by any natural number
greater than n.
As an appendix, we discuss the case where the screen Y is a (combinatorial) tree and
give some answer to Exercise of Section 31.32 in [11]. Precisely, we prove the following
2
proposition:
Proposition 1.8. Assume that a sequence (X ,d ,µ ) of mm-spaces is a L´evy
{ n n n }∞n=1
family. Then, we have
Lip
sup diam(X 1 T,m κ) T is a tree 0 as n
n n
{ −→ − | } → → ∞
for any κ > 0.
5
2. Preliminaries
Let (X,d) be a metric space. For x X, r > 0, and A,B X, we put
∈ ⊆
B (x,r) := y X d(x,y) r , d(A,B) := inf d(a,b) a A,b B ,
X
{ ∈ | ≤ } { | ∈ ∈ }
d(x,A) := d( x ,A), A := y X d(y,A) r , A := y X d(y,A) < r .
r +r
{ } { ∈ | ≤ } { ∈ | }
We denote by m and m the total measure of mm-spaces X and X respectively, and by
n n
Suppµ the support of a Borel measure µ.
2.1. Observable diameter and L´evy radius. In this subsection, we prove several
results in [11] because we find no proof anywhere. Let (X,d,µ) be an mm-space and
f : X R a Borel measurable function. A number a R is called a pre-L´evy mean of
0
→ ∈
f if f (µ) ( ,a ] m/2 and f (µ) [a ,+ ) m/2. We remark that a does exist,
0 0 0
∗ −∞ ≥ ∗ ∞ ≥
but it is not unique for f in general. Let A R be the set of all pre-L´evy means of f.
f
(cid:0) (cid:1) (cid:0) ⊆ (cid:1)
The proof of the following lemma is easy and we omit the proof.
Lemma 2.1 (cf. [11, Section 31.19]). A is a closed bounded interval.
2 f
The L´evy mean m of f is defined by m := (a + b )/2, where a := minA and
f f f f f f
b := maxA . For κ > 0, we define the L´evy radius, LeRad(X; κ), as the infimum of
f f
−
ρ > 0 such that every 1-Lipschitz function f : X R satisfies µ( x X f(x) m
f
→ { ∈ | | − | ≥
ρ ) κ.
} ≤
Lemma 2.2 (cf. [11, Section 31.32]). For any κ > 0, we have
2
Lip
diam(X 1 R,m κ) 2LeRad(X; κ).
−→ − ≤ −
Proof. Put ρ := LeRad(X; κ). It follows from the definition of the L´evy radius that
−
µ( x X f(x) m ρ ) κ for any 1-Lipschitz function f : X R. So we obtain
f
{ ∈ | | − | ≥ } ≤ →
f (µ) (m ρ,m +ρ) m κ, which implies diam(f (µ),m κ) diam(m ρ,m +
f f f f
ρ∗) = 2ρ. T−his completes≥the p−roof. ∗ − ≤ − (cid:3)
(cid:0) (cid:1)
Lemma 2.3 (cf. [11, Section 31.32]). For any κ with 0 < κ < m/2 we have
2
Lip
LeRad(X; κ) diam(X 1 R,m κ).
− ≤ −→ −
Lip
Proof. Put a := diam(X 1 R,m κ). For any ε > 0, there exists a closed interval
−→ −
X R such that f (µ)(X ) m κ and diamX a+ε. We shall show that m X .
0 0 0 f 0
⊆ ∗ ≥ − ≤ ∈
If X ( ,m ), we have
0 f
⊆ −∞
m m
< m κ f (µ)(X ) f (µ) ( ,m ) ,
0 f
2 − ≤ ∗ ≤ ∗ −∞ ≤ 2
which is a contradiction. In the same way, we ha(cid:0)ve X * (m(cid:1) ,+ ). Hence, we get
0 f
∞
m X , which yields X [m (a+ε),m +(a+ε)] since diamX a+ε. Therefore,
f 0 0 f f 0
∈ ⊆ − ≤
we obtain
µ( x X f(x) m a+ε ) = f (µ)([m (a+ε),m +(a+ε)])
f f f
{ ∈ | | − | ≤ } ∗ −
f (µ)(X ) m κ.
0
≥ ∗ ≥ −
6 KEI FUNANO
As a result, we have LeRad(X; κ) a+ε, which completes the proof of the lemma. (cid:3)
− ≤
Combining Lemma 2.2 with Lemma 2.3, we obtain the following corollary:
Corollary 2.4 (cf. [11, Section 31.32]). A sequence X of mm-spaces is a L´evy
2 { n}∞n=1
family if and only if LeRad(X ; κ) 0 as n for any κ > 0.
n
− → → ∞
2.2. L´evy radius and concentration function. Given an mm-space (X,d,µ), we de-
fine the function α : (0,+ ) R by
X
∞ →
α (r) := sup µ(X A ) A is a Borel subset of X such that µ(A) m/2 ,
X +r
{ \ | ≥ }
and call it the concentration function of X. Although the following lemmas and corollary
are somewhat standard, we prove them for the completeness of this paper.
Lemma 2.5 (cf. [14, Section 1.3]). For any r > 0 we have
LeRad(X; 2α (r)) r.
X
− ≤
Proof. Let f : X R be a 1-Lipschitz function. We put A := x X f(x) m and
f
→ { ∈ | ≤ }
A := x X m f(x) . Then,
′ f
{ ∈ | ≤ }
x X f(x) m +r = X x X f(x) < m +r X A .
f f +r
{ ∈ | ≥ } \{ ∈ | } ⊆ \
In the same way,
x X m f(x)+r X A .
{ ∈ | f ≥ } ⊆ \ ′+r
Since µ(A) m/2 and µ(A) m/2, we have
′
≥ ≥
µ( x X f(x) m r ) = µ( x X f(x) m +r )+µ( x X m f(x)+r )
f f f
{ ∈ | | − | ≥ } { ∈ | ≥ } { ∈ | ≥ }
µ(X A )+µ(X A )
≤ \ +r \ ′+r
2α (r).
X
≤
(cid:3)
This completes the proof.
Lemma 2.6 (cf. [14, Section 1.3]). For any κ with 0 < κ < m/2, we have
α 2LeRad(X; κ) κ.
X
− ≤
Proof. Let A be a Borel subset o(cid:0)f X such that µ(cid:1)(A) m/2. We define a function
≥
f : X R by f(x) := d(x,A). Putting ρ := LeRad(X; κ), by the definition of the L´evy
→ −
radius, we have µ x X f(x) m ρ κ. Then we get
f
{ ∈ | | − | ≥ } ≤
µ x X f(cid:0)(x) mf < ρ A µ (cid:1)x X f(x) mf < ρ +µ(A) m
{ ∈ | | − | }∩ ≥ { ∈ | | − | } −
(m κ)+m/2 m = m/2 κ > 0.
(cid:0) (cid:1) (cid:0) (cid:1)
≥ − − −
Hence, there exists a point x x X f(x) m < ρ A and we have m =
0 f f
∈ { ∈ | | − | } ∩
f(x ) m < ρ. Consequently, we obtain
0 f
| − |
µ(X A ) = µ x X f(x) 2ρ µ x X f(x) m ρ κ,
+2ρ f
\ { ∈ | ≥ } ≤ { ∈ | | − | ≥ } ≤
(cid:3)
which completes the p(cid:0)roof of the lemma. (cid:1) (cid:0) (cid:1)
7
Corollary 2.7. Asequence X of mm-spacesis a L´evyfamilyif andonlyif α (r)
{ n}∞n=1 Xn →
0 as n for any r > 0.
→ ∞
Proof. Let X be a L´evy family. Fix r > 0 and take any ε > 0. For an n N with
{ n}∞n=1 ∈
m /2 ε, we have α (r) ε. Hence, we only consider the case of m /2 > ε. From the
n ≤ Xn ≤ n
assumption, we have 2LeRad(X ; ε) r for any sufficiently large n N. Therefore, by
n
− ≤ ∈
virtue of Lemma 2.6, we have
α (r) α 2LeRad(X ; ε) ε,
Xn ≤ Xn n − ≤
which shows αXn(r) → 0 as n → ∞. (cid:0) (cid:1)
Conversely, assume that α (r) 0 as n for any r > 0. Fix κ > 0 and take
Xn → → ∞
any ε > 0. From the assumption, we have 2α (ε) κ for any sufficiently large n N.
Xn ≤ ∈
Therefore, applying Lemma 2.5 to X , we obtain
n
LeRad(X ; κ) LeRad X ; 2α (ε) ε.
n − ≤ n − Xn ≤
(cid:3)
This completes the proof. (cid:0) (cid:1)
2.3. Concentration function and separation distance. Let (X,d,µ) be an mm-
space. For any κ ,κ , ,κ R, we define
0 1 N
··· ∈
Sep(X;κ , ,κ ) = Sep(µ;κ , ,κ )
0 N 0 N
··· ···
:= sup mind(X ,X ) X , ,X are Borel subsets of X
i j 0 N
{ i=j | ···
6
which satisfy µ(X ) κ for any i ,
i i
≥ }
and call it the separation distance of X. In this subsection, we investigate relationships
betweentheconcentrationfunctionandtheseparationdistance. Theproofofthefollowing
lemma is easy, and we omit the proof.
Lemma 2.8 (cf. [11, Section 31.33]). Let (X,d ,µ ) and (Y,d ,µ ) be two mm-spaces.
2 X X Y Y
Assume that a 1-Lipschitz map f : X Y satisfies f (µ ) = µ . Then we have
X Y
→ ∗
Sep(Y;κ , ,κ ) Sep(X;κ , ,κ ).
0 N 0 N
··· ≤ ···
Let us recall that the Hausdorff distance between two bounded closed subsets A and B
in a metric space X is defined by
d (A,B) := inf ε > 0 A B , B A .
H +ε +ε
{ | ⊆ ⊆ }
It is easy to check that d is the metric on the set of all bounded closed subsets of X.
H X
C
Lemma 2.9 (Blaschke, cf. [2, Theorem 4.4.15]). If X is a compact metric space, then
( ,d ) is also compact.
X H
C
Lemma 2.10. Let (X,d,µ) be an mm-space and assume that Suppµ is connected. Then,
for any r > 0 with α (r) > 0 we have
X
m
Sep X; ,α (r) r.
X
2 ≤
(cid:16) (cid:17)
8 KEI FUNANO
Proof. The proof is by contradiction. We may assume that X = Suppµ. Suppose that
Sep X;m/2,α (r) > r, there exist r > 0 with r > r and Borel subsets X ,X X
X 0 0 1 2
⊆
such that µ(X ) m/2, µ(X ) α (r), and d(X ,X ) > r . Let us show that (X ) +
1 2 X 1 2 0 1 r
(cid:0) ≥ (cid:1) ≥
(X ) . If (X ) = (X ) , we have X = (X ) X (X ) . Since X is connected,
1 +r0 1 r 1 +r0 1 r ∪ \ 1 +r0
we get either X = or X (X ) = . It follows from µ (X ) µ(X ) m/2 > 0
1 ∅ \ 1 +r0 ∅ (cid:0) 1(cid:1)r ≥ 1 ≥
that (X ) = . By d(X ,X ) > r , we obtain X X (X ) , which implies that
1 r 6 ∅ 1 2 0 2 ⊆ \(cid:0) 1 +(cid:1)r0
µ X (X ) µ(X ) α (r) > 0. Therefore, we have X (X ) = , which is
\ 1 +r0 ≥ 2 ≥ X \ 1 +r0 6 ∅
a contradiction. Thus, there exists a point x X (X ) X (X ) . Taking a
(cid:0) (cid:1) 0 ∈ \ 1 r \ \ 1 +r0
sufficiently smallball B centered atx such that B X (X ) andB X (X ) = ,
0 (cid:0)⊆ \ 1(cid:1)r (cid:0) ∩ \ (cid:1)1 +r0 ∅
we have
(cid:0) (cid:1)
µ(X ) α (r) µ X (X )
2 X 1 r
≥ ≥ \
µ B (X (X ) ) = µ(B)+µ X (X )
≥ (cid:0) ∪ \ (cid:1) 1 +r0 \ 1 +r0
> µ X (X ) µ(X ),
(cid:0) \ 1 +r0 ≥ (cid:1) 2 (cid:0) (cid:1)
(cid:3)
which is a contradiction. Therefore, we have finished the proof.
(cid:0) (cid:1)
Remark 2.11. If Suppµ is disconnected, the above lemma does not hold in general. For
example, consider the space X := x ,x with a metric d given by d(x ,x ) := 1 and
1 2 1 2
{ }
with a Borel probability measure µ given by µ( x ) = µ( x ) := 1/2. In this case, we
1 2
{ } { }
have α (1/2) = 1/2 and Sep(µ,1/2,1/2) = 1.
X
Lemma 2.12. For any r > 0 there exists a Borel subset X X such that
0
⊆
m
µ X (X ) = α (r) and µ(X ) .
0 +r X 0
\ ≥ 2
Proof. From the definitio(cid:0)n of the con(cid:1)centration function, for any n N, there exist a
∈
closed subset A X such that
n
⊆
m 1
µ(A ) and µ X (A ) + α (r).
n n +r X
≥ 2 \ n ≥
Take an increasing sequence K K (cid:0) of comp(cid:1)act subsets of X such that µ(K )
1 2 n
⊆ ⊆ ··· →
m asn . By using Lemma 2.9 andthe diagonalargument, we have that A K
→ ∞ { n∩ i}∞n=1
Hausdorff converges to a closed subset B K and X (A ) K Hausdorff
i ⊆ i { \ n +r ∩ i}∞n=1
converges to a closed subset Ci ⊆ Ki for each i ∈ N. (cid:0)Put K := (cid:1)∞i=1Ai,K := ∞i=1Ci,
X := K, and Y := K. It is easy to check that B B and C C . We
0 0 1 2 S 1 2 S
⊆ ⊆ ··· ⊆ e ⊆ ···
will show that d(K,K) r by contradiction. If d(K,K) < r, there exists r > 0 such
0
≥
that d(K,K) < r <er. Hence there exist x K and y K such that d(x,y) < r .
0 0
∈ ∈
There exists i N suech that x B and y C , becaeuse both B and C
∈ ∈ i ∈ i { n}∞n=1 { n}∞n=1
are increasieng sequences. Since both A K and X (Ae ) K Hausdorff
{ n∩ i}∞i=1 { \ n +r ∩ i}∞n=1
converge to B and C respectively, there exist two sequences x , y X
i i (cid:0) { n(cid:1)}∞n=1 { n}∞n=1 ⊆
such that d(x ,x),d(y ,y) 0 as n , and x A ,d(y ,A ) r for any n N.
n n n n n n
→ → ∞ ∈ ≥ ∈
Therefore, for any sufficiently large n N we have
∈
d(x ,y ) d(x ,x)+d(x,y)+d(y ,y) < r +d(x ,x)+d(y ,y) < r,
n n n n 0 n n
≤
9
which is a contradiction, because x A and d(y ,A ) r. Thus, we obtain d(K,K)
n n n n
∈ ≥ ≥
r whichyieldsthatd(X ,Y ) r andthereforeY X (X ) . Letusshowthatµ(X )
0 0 0 0 +r 0
≥ ⊆ \ ≥
m/2 and µ(Y ) α (r). For any ε > 0 there exists n N such that µ(K )+ε e m.
0 ≥ X 0 ∈ n0 ≥
Take any δ > 0. Then, for any sufficiently large m we have A K (B ) . Therefore,
m∩ n0 ⊆ n0 δ
m
µ (X ) µ (B ) µ(A K ) µ(A )+µ(K ) m ε.
0 δ ≥ n0 δ ≥ m ∩ n0 ≥ m n0 − ≥ 2 −
By taking(cid:0)δ 0(cid:1), we (cid:0)obtain µ(cid:1)(X ) m/2 ε, which shows µ(X ) m/2. In the same
0 0
way, we have→µ(Y ) α (r). This c≥omplete−s the proof. ≥ (cid:3)
0 X
≥
Lemma 2.12 directly implies
Lemma 2.13. For an mm-space X and r > 0, we have
m
Sep X; ,α (r) r.
X
2 ≥
(cid:16) (cid:17)
Corollary 2.14. If a sequence X of mm-spaces satisfies that Sep(X ;κ,κ) 0 as
{ n}∞n=1 n →
n for any κ > 0, we then have α (r) 0 as n for any r > 0.
→ ∞ Xn → → ∞
Proof. Suppose that there exists c > 0 such that α (r) c for infinitely many n N.
Xn ≥ ∈
Applying Lemma 2.13 to X , we have
n
m
n
r Sep X ; ,α (r) Sep X ;α (r),α (r) Sep(X ;c,c).
≤ n 2 Xn ≤ n Xn Xn ≤ n
(cid:16) (cid:17)
This is a contradiction, since the right-han(cid:0)d side of the above(cid:1)inequality converges to 0
as n . This completes the proof. (cid:3)
→ ∞
Lemma 2.15 (cf. [14, Lemma 1.1]). Let (X,d,µ) be an mm-space. Assume that a Borel
subset A X and r > 0 satisfy µ(A) κ and α (r ) < κ. Then, for any r > 0 we have
0 X 0
⊆ ≥
µ X A α (r).
\ +(r0+r) ≤ X
Corollary 2.16. Assumethat as(cid:0)equence (X ,(cid:1)d ,µ ) ofmm-spacessatisfyα (r)
{ n n n }∞n=1 Xn →
0 as n for any r > 0. Then, we have Sep(X ;κ,κ) 0 as n for any κ > 0.
n
→ ∞ → → ∞
Proof. Since Sep(X ;κ,κ) = 0 for n N with m < κ, we assume that m κ for any
n n n
∈ ≥
n N. For any ε > 0, we have α (ε) < κ/2 for any sufficiently large n N from the
∈ Xn ∈
assumption. Thus, it follows from Lemma 2.15 that µ X (A ) α (ε) < κ/2
n n \ n +2ε ≤ Xn
for any Borel sets A ,B X with µ (A ),µ (B ) κ. In the same way, we get
n n n n n n n
⊆ (cid:0)≥ (cid:1)
µ X (B ) < κ/2. Therefore, we obtain
n n n +2ε
\
(cid:0) µn Xn (cid:1) (An)+2ε (Bn)+2ε µn Xn (An)+2ε +µn Xn (Bn)+2ε
\ ∩ ≤ \ \
< κ m ,
(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) n (cid:1) (cid:0) (cid:1)
≤
which implies µ (A ) (B ) > 0, and thereby (A ) (B ) = . As a con-
n n +2ε n +2ε n +2ε n +2ε
∩ ∩ 6 ∅
sequence, we have diam(A ,B ) 4ε, which shows Sep(X ;κ,κ) 4ε. This completes
n n n
(cid:0) ≤ (cid:1) ≤ (cid:3)
the proof.
Combining Corollary 2.14 with Corollary 2.16, we obtain the following corollary:
10 KEI FUNANO
Corollary 2.17 (cf. [11, Section 31.33]). A sequence X of mm-spaces is a L´evy
2 { n}∞n=1
family if and only if Sep(X ;κ,κ) 0 as n for any κ > 0.
n
→ → ∞
2.4. Lp-concentrationof maps. Inthissubsection, weinvestigaterelationshipsbetween
concentration of maps and Lp-concentration of maps. The standard argument yields the
following basic fact.
Lemma 2.18. For 0 < p < q, Lq-concentration of maps implies Lp-concentration of maps
under the assumption supm < + .
n
n N ∞
∈
Lemma 2.19. Let X be an mm-space and Y a metric space. For any κ > 0, p (0,+ ],
∈ ∞
and a Borel measurable map f : X Y, we have
→
2
diam(f (µ ),m κ) V (f).
∗ X − ≤ (κm)1/p p
Proof. The case of p = is easy, so we consider the case of p < + only. Assume that
∞ ∞
µ x X d f(x),f(x) ε > κ for any x X and ε := V (f)/(κm)1/p. By
X Y ′ 0 ′ 0 p
∈ | ≥ ∈
the Chebyshev’s inequality, we get
(cid:0)(cid:8) (cid:0) (cid:1) (cid:9)(cid:1)
Vp(f)p = dY f(x),f(x′) p dµX(x) dµX(x′) > εp0κ dµX(x′) = εp0κm = Vp(f)p.
ZX nZX o ZX
(cid:0) (cid:1)
Hence, there is a point x X such that µ x X d f(x),f(x) ε κ. This
′ X Y ′ 0
∈ ∈ | ≥ ≤ (cid:3)
completes the proof.
(cid:0)(cid:8) (cid:0) (cid:1) (cid:9)(cid:1)
Corollary 2.20. For any p (0,+ ], Lp-concentration of maps implies concentration
∈ ∞
of maps.
Lemma 2.21. Let X be an mm-space with a finite diameter and Y a metric space. Then,
for any κ > 0, p (0,+ ), and 1-Lipschitz map f : X Y, we have
∈ ∞ →
κ p κ
V (f)p m2diam f (µ ),m + 2m κ.
p ≤ ∗ X − (diamX)p − (diamX)p
(cid:16) (cid:17) (cid:16) (cid:17)
Proof. For any ε > 0 with diam f (µ ),m κ/(diamX)p < ε, there exists a Borel
X
subset A Y such that diamA < ε∗and f (µ−)(A) m κ/(diamX)p. By diamA < ε,
X
⊆ (cid:0) ∗ ≥ − (cid:1)
we get
(2.1) dY f(x),f(x′) p dµX(x)dµX(x′) m2εp.
≤
Z Zf−1(A) f−1(A)
×
(cid:0) (cid:1)
Since
κ 2
(µ µ ) X X f 1(A) f 1(A) m2 m
X × X × \ − × − ≤ − − (diamX)p
(cid:0) (cid:1) (cid:16) κ (cid:17)κ
= 2m ,
− (diamX)p (diamX)p
(cid:16) (cid:17)