109
Revista de la
Union Matematica Argentina
Volumen 42, Nro . 1, 2000, 109- 1 1 2
D OUBLING PROPERTY FOR THE HAAR MEASURE
ON QUASI-METRIC GROUPS
Hugo Aimar (*) and Bibiana Iaffei ( * * )
1
( * ) Depto.de Matematica, FIQ-UNL. PEMA-INTEC. Giiemes 3450. (3000) Santa
Fe, Argentina. E-mailaddress: [email protected].
( * * ) Depto. de Matematica, FaFoDoc-VNL. Paraje El Pozo. (3000) Santa Fe, Ar­
gentina. E-mailaddress: [email protected].
Abstract: In [L-S] , [V-Kl] and [V-K2L doubling measures on doubling complete met­
ric spaces are constructed, . solving a problem posed by Dyn'kin [D] . In [M] a metric
on self-similar fractals is defined in such a way that Hausdorff measure is doubling,
in other words, for self-similar fractals the problem of Dyn'kin is solved by the nat­
ural measure for the underlying fractal structure. In this note we show that Haar
measure solves Dyn'kin's conjecture on complete quasi-metric doubling groups.
l .Introduction
In a recent paper by Luukkainen and Saksman [L-S] , based on the previous work
of Vol'berg and Konyagin, [V-Kl] and [V-K2J , a non trivial doubling measure is
constructed on each complete metric space with the doubling or weak homogene­
ity property: there exists a constant N such that any subset A of a ball B{x, r)
with the property d(y, z ) 2: r /2, Y E A, z E A, y '" z; has at most N points.
This weak homogenity was first qbserved by Coifman and Weiss, [C-W] , as a nec­
essary condition in a space of homogeneous type: (X, d, J.L) is a space of homo­
geneous type if d is a non-negative, symmetric and faithfull function defined on
X x X , satisfying for some K 2: 1 the quasi-triangle inequality d(x, z ) ::; K {d(x , y) +
. d{y, z )) , x, y, z E X ; J.L is a doubling measure defined on a a-algebra containing the
d-balls, 0 < J.L { B {x , 2r) ) ::; A J.L { B {x, r)) < 00, for some A, every x E X and every
r > O.
In the euclidean space Rn , the tipical space of homogeneous type with its standard
metric and measure, Lebesgue measure is at once Hausdorff measure of dimension n
and Haar measure for the underlying group structure. In a recent paper, U. Mosco
[M] proves that any self-similar compact fractal K, in the sense of Hutchinson, with
the Hausdorff measure of the apropriate dimension carries a quasi-distance such that
K becomes a space of homogeneous type. In this note we consider the doubling
property for Haar measures on groups. We show that for a complete abelian group
X, whose topology is given by a translation invariant quasi-distance with the weak
homogeneity property, (X, d, J.L) is a space of homogeneous type, if J.L is the Haar
measure on X.
1 1991 Mathematics Subject Classification 28A12.
CONICET, Argentina.
Supported by Progr.
CAl+D UNL­
110
Doubling property for the Haar measures . . ;
2. Doubling
Haar measures
As shown by Macias and Segovia [M�SJ , every quasi":metric space is metrizable in
the sense that there exist a distance p and a positive number 0: such . that pQ is
equivalent to d. So that, without loosing generality, we shall assume that d-balls
are open sets. We shall follow closely the developement and notation in [K- TJ , page
2 54 for the construction of the Haar measure. Let us write Br for the d-ball with
radius r > 0 centered at the identity 0 of X.
Let Co denote the class of all bounded and open non-empty subsets of X. Given
HI and H2 in Co , the minimal number (HI : H2) of translates of H2 that are needed
to cover HI is finite. This follows from the w�ak homogeneity property, since HI is
contained in a ball and H2 contains a ball. Of course we also have (HI : H2) � 1
and (HI : H2) = ( x + HI : H2) for every x E X. Moreover, for HI , H2 and H3 in
Co, we have
(2. 1)
For a given n E N, the expression: An ( G) � (G : BI/n) (B 1 : BI/n ) - l , defines a· finite
set function on the class Co , which from (2. 1) and the weak homogeneity property
satisfies a basic uniform doubling condition as shown in the next lemma.
Lemma
2.2. The set functions An satisfy uniform doubling conditions:
An (B(x, 2r) ) :::; AAn (B( x, r) ) ,
for every x E X, every r > 0 and some constant A bounded b y the weak homogeneity
constant of (X, d) .
Proof. Take HI = B ( x, 2r ) , H2 = B(x, r) and H3 = BI /n in (2. 1) , then
( B (x, 2r ) : Bl/n) :::; (B (x , 2r) : B(x, r)) ( B (x, r) : B1 /n) .
(2 . 3)
Notice now that ( B (x , 2r ) : B(x, r ) ) = (B2r : Br) , the number of translates of Br
needed to cover B2r is bounded by the weak homogeneity constant N . Dividing
both sides of (2.3) by (BI : BI/n ) we. obtain the result.
D
Next we apply a Hahn-Banach extension argument to produce a content A, as a
generalized limit of An , satisfying all the relevant properties of each An'
We have a set function A on Co satisfying the following properties
0 < (B::G) :::; A(G) :::; (G : Bl ) < 00 j
A(G1 U G2 ) :::; A(Gl) + A(G2) j
A(Gl U G2) = A(Gl ) + A(G2) if d(Gl , G2) > 0 j
A(Gt} :::; A(G2) if Gl C G2 j
A(X + G) A(G) j
for some constant A less than or equal to N, the inequalities
Lemma 2.4.
(2.5)
(2.6)
(2 .7)
(2.8)
(2.9)
(2. 10 )
=
0 < A(B(x , 2r)) :::; AA{ B(x , r )) <
hold for every x E X and every
r >
O.
00
Hugo Aimar and Bibiana Iaffei
The last step
is
the
use
f.L* ( E )
=
111
of the content
inf {
00
I: >'(G; )
;=1
:
>. t o construct the metric outer measure
Gi
E Co U {0 , X } ;
00
U G;
;=1
�
E} ,
with >.( <p) = 0 and >'(X) = +00 if X is unbounded. The a-algebra of f.L* -measurable
subsets of X contains the Borel sets and, in particular, d-balls are f.L* -measurable
sets. Let us check that f.L* satisfies the doubling property. Observe that, since
f.L* (B) � >.(B) � (B : B 1 ) , every ball has finite measure. Since bounded sets are
totally bounded sets in quasi-metric spaces with the weak homogeneity property, if
follows from the completeness that the closure of a ball is a compact set. Then if
B(x, s)
c
00
U Gj , with Gj E Co , B (x, s/2) and also B (x, s/2) can be covered by only
j =l
a finite number of the G/ s
must have that
.
Say B(x, s /2 )
C
J
U Gj • From (2. 10) and (2.6) we
j =l
0 < >'(B (x, s)) � A >'( B (x, s/2)}
�
�
00
J
A 2: >'(Gj )
j= 1
A 2: >' (Gj)
j =1
00
In other words, >'(B (x, s)) � A 2: >'(Gj) for every covering {Gj } of B(x, s).
j =l
that
0 < >'(B(x, s)) � AJL* (B(x, s) ) ,
for every x E X and every s > O.
Thus
0 < f.L* (B(x, 2r)) � >'(B(x, 2r))
� A>.(B (x, r))
� A2f.L* (B(x, r)) <
00 .
0
So
112
Doubling prop�rty for the . Haar measures . . .
References
[ C-W]
[D ]
[K-T]
[1-S]
[M]
[M-S]
[V-Kl]
[V-K2]
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tive sur certains espaces homogenes ", Lecture Notes in Math. , Vol.
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Dyn'kin, E. M., "Problem 8. 1 1 in Linear and Complex Analysis
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1984.
Kingman, J. F. C. and Taylor, S. J . , "Introduction to measure and
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Luukkainen, J . and Saksman, E . , "Every complete doubling metric
space carries a doubling measure", Proceedings of the AMS , Vol.
126, Nro. 2, 1998, 531� 534.
Mosco, U . , "Lagrangian metrics on !mctals ", Proceedings of Sym­
posia in Applied Mathematics, 54, 1998, 301-323.
Ma.cias, R. and Segovia, C . , "Lipschitz functions on spaces 0/ ho­
mogeneous type ", Advances in Math. 33, 1979, 257-270.
Vol'berg, A. L. and Konyagin, S. V., "There is a homogeneous
measure on any compact subset in Rn» , Soviet Math. Dokl. 30,
1984, 453-456.
, "On measures with the doubling condition ",
Math. USSR-Izv. 30, 1988, 629-638.
Recibido
24
de . Marzo
de
2000
Aceptado
29
de
de
2001
Marzo