Monatsh. Math. 136, 143±150 (2002)
On the Box Dimension of Typical Measures By 1
Jozef Myjak and Ryszard Rudnicki2 2
1 Universita di L'Aquila, L'Aquila, Italy Polish Academy of Sciences, Katowice, Poland
Received 29 November 2000; in ®nal form 8 January 2002 Abstract. Let
X; be a complete metric space and let M be the space of all probability Borel measures on X. We give some estimations of the upper and lower box dimensions of the typical (in the sense of Baire category) measure in M. 2000 Mathematics Subject Classi®cation: 28A78; 26A21, 54E52 Key words: Box dimension, measure, residual subset
1. Introduction In the analytic theory of the dimension of sets and measures one of the most frequently used notions is the box dimension (called also Minkowski or entropy dimension). For some recent results concerning box dimension see [1, 5, 6±8, 13, 17]. The early bibliographical references can be found in the monographs [2, 9, 14]. In this note we study some typical properties of the box dimension of measures. Recall that a subset of a metric space is called of the ®rst Baire category, if it can be represented as a countable union of nowhere dense sets. A subset of a complete metric space X is called residual, if its complement is of the ®rst Baire category. If the set of all elements of X satisfying some property P is residual in X, then property P is called generic or typical. We also say that the typical elements of X has property P. The typical properties of dimensions of sets have been investigated in [3, 5, 10, 11, 16]. Some typical (but in the sense of prevalence) properties of dimensions of measures were established in [15]. While, the typical properties of measures in the sense of Baire category were studied in [4]. More precisely, Genyuk proved that for a typical probability measure on a Polish space X, the lower local dimension is equal to 0 and the upper local dimension is equal to 1, for all x except a set of ®rst category. (Here by lower=upper local dimension of a measure we mean the lower=upper limit of log
B
x; r=logr as r ! 0, see [12].) Further, she shows that the Hausdorff dimension of a typical measure is equal to 0. In this note we prove that for a typical probability measure de®ned on a complete metric space, This research was partially supported by the State Committee for Scienti®c Research (Poland) Grant No. 2 P03A 010 16 (RR). This work was partially written while R.R. was a visitor to the University of L'Aquila. The authors thank the group GNAFA (CNR) for ®nancial support of this visit.
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J. Myjak and R. Rudnicki
the lower box dimension is equal to 0 whereas the upper box dimension is greater than or equal to the smallest local upper box dimension of X. Moreover, we show that for a typical probability measure on a complete separable metric space its support is equal to the whole space. 2. Notation Let
X; be a complete metric space and let B
x; r denote the closed ball in X with centre at x and radius r > 0. By B
X we denote the -algebra of Borel subsets of X and by M we denote the set of all probability Borel measures on X. As usual, for A X the symbol A stands for the closure of A, the symbol card A stands for the cardinality of A and for x 2 X the symbol x stands for the delta Dirac measure supported at x. Throughout this paper we assume that the space M is endowed with the FortetMourier distance denoted dist and given by the formulae
dist
1 ; 2 sup f
x 1
dx f
x 2
dx : f 2 L ;
1 X
X
for 1 ; 2 2 M. Here L is the subset of C
X which contains all the functions f such that j f
xj 4 1 and j f
x f
yj 4
x; y for x; y 2 X:
2 It can be proved that the sequence
n , n 2 M, is weakly convergent to a measure 2 M, if and only if lim dist
n ; 0: n!1
It is well known that the space M endowed with the metric dist is complete. We recall that the lower and upper box dimensions of a set E X are de®ned, respectively, by the formulae log N
E; r ;
3 dim E lim inf r!0 log
1=r log N
E; r ;
4 log
1=r where N
E; r is the least number of closed balls of radius r which cover the set E. Note that if the closure of E is non-compact then dim E dim E 1. Note also that lower and upper box dimensions of a set are invariant under topological closure. dim E lim sup r ! 0
Remark 1. In the de®nition of box dimensions we can replace the number N
E; r by M
E; r ± the greatest possible number of disjoint closed balls of radius r that may be found with centres in E. Let 2 M. The quantities dim lim inffdim E : E 2 B
X;
E 5 1 g
5 !0
and dim lim inffdim E : E 2 B
X;
E 5 1 !0
g
are called the lower and upper box dimension of , respectively.
6
On the Box Dimension of Typical Measures
145
3. Results Theorem 1. The set f 2 M : dim 0g is residual in the space M. Proof. Denote by F the set of all measures 2 M with ®nite supports. For 2 F we de®ne Dn
f 2 M : dist
; < 3 where k card
supp . Let D
1 \
Dn
[
where Dn
n1
k n
g;
Dn
:
2F
Since for each n 2 N the set Dn is open and dense in the space M, the set D is residual. To complete the proof it is suf®cient to show that dim 0 for every 2 D. Let 2 D be given. Since 2 D, for every n 2 N there exists a measure n 2 F such that dist
; n < 3
k
n n
;
7
where k
n card An
and An supp n :
Now de®ne k
n n
En fx 2 X :
x; An 4 2 Observe that
En > 1 Indeed, let f
x maxf2
k
n n
52
n 2 : 3
8
x; An ; 0g. Since f 2 L we have
f
x dn
x f
x d
x
dist
; n 5
X k
n n
X
n
An
2
k
n n
By (9) and (7) we receive
En 5 n
An
g:
k
nn
2
dist
; n > 1
En :
9
k
nn 2 ; 3
whence (8) follows immediately. Let > 0 be given. Using (8) one T can easily verify that there exists n0 2 N such that
E 5 1 , where E 1 nn0 En . We check that dim E 0. Indeed, set "n 2 k
n n . Let An fx1 ; . . . ; xk
n g. Since En
k
n [ i1
B
xi ; 2
k
n n
146
J. Myjak and R. Rudnicki
and E En for n 5 n0, we have N
E; "n 4 k
n for n 5 n0. Thus we have 0 4 dim E 4 lim
n!1
log N
E; "n log k
n 0: 4 lim n ! 1
n k
nlog 2 log
1="n
From this and (5) it follows that dim 0. This completes the proof.
&
Now we will formulate a result concerning the upper box dimension of the typical measure. Let 0 denote the smallest local upper box dimension of X, i.e. the quantity given by the formula 0 0
X inffdim B
x; r : x 2 X; r > 0g:
10
Theorem 2. The set f 2 M : inffdim E :
E > 0g 5 0 g: is residual in M. Proof. Without any loss of generality we can assume that 0 > 0. Fix 2
0; 0 and > 0. Let x 2 X and r; " > 0. By M
x; r; " we denote the greatest possible number of disjoint closed balls of radius " that may be found with centres in the ball B
x; r. Since dim B
x; r > , from (3) and Remark 1 it follows that there is " 2
0; 1 such that M
x; r; " 5 " :
11
For every x 2 X and r > 0 we ®x an " satisfying (11) and we will denote it by "
x; r. It is obvious that "
x; r < 2r. Now, in the ball B
x; r we choose the points y1 ; . . . ; yM , where M M
x; r; "
x; r, such that B
yi ; "
x; r \ B
yj ; "
x; r ; for
i; j 2 f1; . . . ; Mg; i 6 j
and we de®ne 1
y yM : M 1 Clearly 2 M. By Fin we denote the family of all ®nite subsets of X. For given A 2 Fin and r > 0, we denote by k
A the number of elements of A and we de®ne 1 X A;r x;r ; k
A x 2 A
A; r min "
x; r; 6 x2A M
A; r f 2 M : dist
; A;r <
A; rg: x;r
Let M
r
[ A 2 Fin
M
A; r
and Gm
1 [ nm
M
1 : n
On the Box Dimension of Typical Measures
147
Since for all n the sets M 1n are open, also the sets Gm are open. Moreover the sets Gm are dense in M. Indeed, since 0 > 0 the space X has P no isolated points. This 1 implies that the set of all measures of the form A k
A , where A 2 Fin, is S1 x 2 A x 1 dense in M. Since dist
A ; A; 1n < n the set Fm nm fA; 1n : A 2 Fing is also dense in M. As Fm Gm , the set Gm is dense in M. Recall that A;r and, consequently, M
r and Gm were constructed for ®xed 2
0; 0 and > 0. Now, for such and we de®ne G
;
1 \
Gm :
12
m1
The set G
; is residual in the space M, because for each m 2 N the set Gm is open and dense in M. Claim. If 2 G
; and E 2 B
X be such that
E 5 then dim E 5 . Indeed, let 2 G
; and E 2 B
X. Since dim E dim E, where E is the closure of E, we can assume that E is a closed set. From (12) it follows that for every m 2 N there exist an integer number n 5 m and a set A 2 Fin such that dist
; A; 1=n <
A; 1n. De®ne 2 1 : Ec fx 2 X :
x; E 4 cg; where c A; n Let f
x maxfc
x; E; 0g. Since f 2 L we have
dist
; A; 1=n 5 f
x d
x f
x dA; 1=n
x 5 c
E X
X
This implies that A; 1=n
Ec 5
E
1 dist
; A; 1=n > c
cA; 1=n
Ec :
1 1 1 A; 5 : c n 2
From the de®nition of the measure A; r it follows that there exists x 2 A such that x; 1=n
Ec > 12 . In turn, this implies that at least 12 M points from the set fy1 ; . . . ; yM g belongs to Ec . Consequently 1 1 1 1 5 M x; ; " x; M Ec ; " x; n 2 n n and by (11) we obtain
1 1 1 M Ec ; " x; 5 " x; : n 2 n
Since c 4 13 " x;
1 n
, we receive 1 1 1 1 5 " x; M E; " x; : 3 n 2 n
13
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J. Myjak and R. Rudnicki
Since " x;
1 n
in (13) can be arbitrary small, we have dim E lim sup " ! 0
log
12 " log M
E; " 5 lim ; "!0 log
" log
"=3
14
which completes the proof of the Claim. Now, let
n be an increasing sequence of positive numbers convergent to 0 and let
n be a decreasing sequence of positive numbers convergent to 0. Then the set G
1 \
G
n ; n
n1
is residual in M and inffdim E :
E > 0g 5 0 for 2 G. This completes the proof.
&
From Theorem 2 follows immediately Corollary 1. For the typical measure in M we have dim 5 0 , where 0 is given by (10). Remark 2. A result similar to Theorem 2 concerning the box and packing dimensions of the typical compact subsets of a complete metric space X were obtained in [11]. It was proved that if the space of all nonempty compact subsets of X is equipped with the Hausdorff metric then the typical compact set has the upper box and packing dimensions at least 0 , where 0 is given by (10). Theorem 3. Assume that
X; is a complete separable metric space. Then the set f 2 M : supp Xg is residual in M. Proof. Let
xn n 2 N be a sequence of elements of X such that for each n0 the subsequence
xn n 5 n0 is dense in X. Let PPbe the set of all real sequences p
pn n 2 N such that pn > P 0 for all n 2 N and 1 n1 pn 1. For each p 2 P we de®ne a measure p by p 1 p . For p 2 P and n 2 N we de®ne n1 n xn 1 min pi ; n 14i4n An
p f 2 M : dist
; p <
p; ng:
p; n
Let An
[ p2P
An
p and A
1 \
An :
n1
Since for each n 2 N the set An is dense and open in the space M, the set A is residual in M. Let 2 A. We check that supp X. Fix a point y 2 X and a
On the Box Dimension of Typical Measures
149
number r 2
0; 1. Let i and n be integers such that
xi ; y < r=2 and n 5 maxfi; 2=rg. Take a measure p 2 An such that dist
; p <
p; n:
15
Since the function f
x maxfr=2
x; xi ; 0g belongs to the set L, we have
r r r B xi ; : f
x d
x 5 p
xi dist
; p 5 f
x dp
x 2 2 2 X X As p
xi pi , the last inequality implies r 5 pi B xi ; 2
2 dist
; p : r
16
Since
p; n 4 1n pi 4 2r pi , inequalities (15) and (16) give r 2 2 5 pi dist
; p > pi
p; n 5 0: B xi ; 2 r r From this and inequality
xi ; y < r=2 it follows that
B
y; r > 0. Since y 2 X and r > 0 were arbitrary, this shows that supp X. The proof is completed. & From Theorem 3 it follows immediately Corollary 2. Let
X; be a complete separable metric space. Then for a typical measure in M we have dim E dim X
and
dim E dim X;
for every E 2 B
X such that
E 1. References [1] Brown G, Yin Q (1997) Box dimension for graphs of fractal functions. Proc Edinburg Math Soc 40: 331±344 [2] Falconer KJ (1997) Techniques in Fractal Geometry. Chichester: Wiley [3] Feng D (1997) Category and dimension of compact subset of Rn . Chinese Sci Bull 42: 1680±1683 [4] Genyuk J (1997=1998) A typical measure typically has no local dimension. Real Anal Exchange 23: 525±537 [5] Gruber PM (1989) Dimension and structure of typical compact sets, continua and curves. Monatsh Math 108: 149±164 [6] Hofbauer F (1996) The box dimension of completely invariant subsets for expanding piecewise monotone transformations. Monatsh Math 121: 199±211 [7] Hu H (1998) Box dimensions and topological pressure for some expanding maps. Comm Math Phys 191: 397±407 [8] Jaffard S (1998) Sur la dimension de boit des graphes. CR Acad Sci Paris Ser I Math 326: 555±560 [9] Mattila P (1995) Geometry of sets and measures in Euclidean spaces. Fractals and recti®ability. Cambridge: Univ Press [10] Myjak J, Rudnicki R (2001) On the typical structure of compact sets. Arch Math 76: 119±126 [11] Myjak J, Rudnicki R (2000) Box and packing dimensions of typical compact sets. Monatsh Math 131: 223±226 [12] Olsen L (1995) A multifractal formalism. Adv in Math 116: 82±196 [13] Orzechowski ME (1998) A lower bound of the box-couting dimension of crossing in fractal percolation. Stochastic Process Appl 74: 53±65 [14] Pesin YB (1997) Dimension Theory in Dynamical Systems. Contemporary Views and Applications. Chicago: Univ of Chicago Press
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J. Myjak and R. Rudnicki: On the Box Dimension of Typical Measures
[15] Sauer T, Yorke JA (1998) Are the dimensions of sets and its image equal under typical smooth functions. Ergodic Theory Dynamical Systems 17: 531±546 [16] Schmeling J, Winkler R (1995) Typical dimension of the graph of certain functions. Monatsh Math 119: 303±320 Authors' addresses: J. Myjak, Dipartimento die Matematica Pura ed Applicata, Universita di L'Aquila, 67100 L'Aquila, Italy, and WMS AGH al. Mick uwicza 30, 30059 Krak ow, Poland, e-mail: myjak@ univaq.it; R. Rudnicki, Institute of Mathematics, Polish Academy of Sciences and Institute of Mathematics, Silesian University, Bankowa 14, 40007 Katowice, Poland, e-mail:
[email protected] 