to verify that it is lower semicontinuous. Hence it follows as above that dimB is of. Baire's class 2. We de ne the packing dimension dimp A for A X by dimp A = inf ...
to appear: Math. Proc. Cambridge Phil. Soc.
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES Pertti Mattila and R. Daniel Mauldin1
(Received 3 January 1995) 1. Introduction
During the past several years, a new type of geometric measure and dimension have been introduced, the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdor measure in that it is de ned in terms of packings rather than coverings. However, in contrast to Hausdor measure, the usual de nition of packing measure requires two limiting procedures, rst the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for de ning packing measure and dimension. In this paper, we nd a basic limitation on this possibility. We do this by determining the descriptive set theoretic complexity of the packing functions. Whereas the Hausdor dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing measure and dimension functions are measurable with respect to the -algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdor measure, e.g., measures of sections and projections, remain true for packing measure. We now give a somewhat more detailed description of our results and we introduce some notation. Throughout this paper (X; d) will be a Polish space, that is, a complete separable metric space. We equip the space K(X ) of non-empty compact subsets of X with the Hausdor distance %;
%(K; L) = supfdist(x; L); dist(y; K ) : x 2 K; y 2 Lg: Then (K(X ); %) is a complete separable metric space, see e.g., [R]. We denote by Hg and Pg the Hausdor and packing measures generated by a non-negative nondecreasing function g on the positive reals; their de nitions will be given later. We shall study the measurability properties of the functions Hg and Pg on K(X ). The Hausdor measure Hg is rather simple in this respect; it is in Baire's class 2, and in particular a Borel function. The packing measure is much more complicated, 1
Research supported by NSF Grant DMS 9303888 1
2
PERTTI MATTILA AND R. DANIEL MAULDIN1
as can be expected from its de nition. Assuming that g satis es the doubling condition; g(2r) kg(r), we show that Pg is measurable with respect to the algebra generated by the analytic sets, but it is not Borel measurable even when X is the unit interval. From these results on measures we obtain that on K(X ) the Hausdor dimension is a Borel function and the packing dimension is measurable with respect to the -algebra generated by the analytic sets. Again, the packing dimension is not a Borel function. We shall actually prove the measurability of the packing dimension in two ways: via the packing measures and via the box counting dimension (also known as the Minkowski dimension). We shall mainly work with packing measures whose de nition is based on the radii of balls. In Section 5 we brie y discuss the measurability questions related to diameter-based packing measures. By composing suitable functions, we can use the above results to deduce measurability of various functions. For example, we show in Section 6 that if K is a compact subset of a product space T X , where T is another Polish space, then the Hausdor measures and dimensions of the sections fx 2 X : (t; x) 2 K g are Borel functions of t, and their packing measures and dimensions are measurable with respect to the -algebra generated by analytic sets. Our example in Section 7 shows that these latter functions need not be Borel measurable. The last section is independent of the others. There we study the lower density ? P A \ B (x; r) g d (A; x) = lim inf ;
g
r!0
g(2r)
where g satis es the doubling condition. For packing measures this is more natural than the corresponding upper density because even for compact subsets K of R n with positive and nite P s measure the upper density may be in nite everywhere in K . For the lower density it was shown in [TT2] and [ST] that if A R n and P s (A) < 1, then dg (A; x) = 1 for P s almost all x 2 A. In Example 8.9 we shall show that this does not extend to any in nite-dimensional inner product space. However, we shall prove for g(r) = rs the optimal inequalities 2?s dg (A; x) 1 for Pg almost x 2 A if A is a subset of an arbitrary Polish space with Pg (A) < 1, see Theorem 8.3. We shall also prove in Theorem 8.5 that dg (A; x) does equal one almost everywhere in A, but almost everywhere with respect to Hg and not Pg . This has the following consequence: if A X with Pg (A) < 1, then Hg (A) = Pg (A) if and only if (1.1)
?
Hg A \ B (x; r) = 1 for P almost x 2 A: lim g r!0 g(2r)
In R n this has been proved in [TT2] and [ST] (see also [M]) with the help of the aforementioned fact that dg (A; x) = 1 for Pg almost all x 2 A. Still in R n , for g(r) = rs (1) implies that s must be an integer and A rather regular, i.e. recti able with respect to Pg . This gives for sets A R n with g(r) = rs and Pg (A) < 1 that Hg (A) = Pg (A) if and only if A is such a recti able set, see [ST] or [M]. We shall extend this result to separable Hilbert spaces in Theorem 8.8.
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES
3
We shall denote by R the set of real numbers, by R n the Euclidean n-space, by Q the set of rational numbers, and by N the set of positive integers. For A X , the closure of A will be A. The -algebra generated by analytic sets will be denoted by B(A). In a product space, proj1 stands for the projection onto the rst factor. Finally, we wish to thank Lars Olsen and Jouni Luukkainen for their corrections and comments during the preparation of this manuscript. 2. Hausdorff measure and dimension functions
Here and later g : [0; 1) ! [0; 1) will be a non-decreasing function with g(0) = 0. We denote by d(A) the diameter of a set A X with d(;) = 0. Let 0 < 1 and A X . The approximating Hausdor g-measure Hg; (A) of A is de ned by
Hg; (A) = inf
X 1
i=1
g(d(Ui)) : A
1 [ i=1
Ui ; each Ui is open with d(Ui) <
and the Hausdor g-measure Hg (A) by
Hg (A) = lim H (A): !0 g; We write Hg; = Hs and Hg = Hs when 0 < s < 1 and g(r) = rs for r 0. The Hausdor dimension dimH A of A is de ned by dimH A = inf fs : Hs (A) = 0g: We shall use the usual notion of Baire's classes for functions between metric spaces. Thus Baire's class 0 consists of all continuous functions and Baire's class n + 1 consists of all pointwise limits of sequences of functions in Baire's class n. In particular, the upper and lower semicontinuous functions belong to the Baire's class 1. 2.1. Theorem. a) For 0 < 1, the function
Hg; : K(X ) ! [0; 1] is upper semicontinuous. b) The functions
Hg and dimH : K(X ) ! [0; 1]
are of Baire's class 2. Moreover, in general, these functions are not of Baire's class 1. Proof. Let V be a countable basis for the topology of X and let fWn g1 n=1 be an enumeration of all nite unions of the sets of V . Let c 2 R and K 2 K(X ). Using the de nition of Hg; and the compactness of K , we see that Hg; (K ) < c if and only if there are nitely many open sets U1 ; : : :; Uk such that k X i=1
g(d(Ui)) < c; K
k [ i=1
Ui and d(Ui ) < for i = 1; : : :; k:
4
PERTTI MATTILA AND R. DANIEL MAULDIN1
This holds if and only if there are n1; : : : ; nm 2 N such that (2.1)
m X i=1
g(d(Wni )) < c; K
m [
i=1
Wni and d(Wni ) < for i = 1; : : :; m:
Since the set of K 2 K(X ) for which there exist n1 ; : : :; nm satisfying (2.1) is clearly open in the exponential or Vietoris topology[K2], we conclude that the set fK 2 K(X ) : Hg; (K ) < cg is open, and a) follows. Letting Q + be the set of positive rationals, it then follows that for each , (2.2)
fF 2 K(X ) : dimH F g =
\ \
s> n s2Q+
fF : H1s=n (F ) < 1g
is a G -set. From this we get fF : < dimH F < g is a G -set, for all 0 < < . Thus, dimH is a Baire class 2 function. As Hg = limn!1 Hg;1=n, part a) completes the proof of the rst part of b). The following example shows that, in general Hg is not a Baire class one function. Consider X = [0; 1] [0; 1] and H1 : K(X ) ! [0; +1]. Since H1 (F ) = 0 for all nite F , H1 has value 0 on a dense subset of K(X ). On the other hand, it is easy to show that each open set of the form P (U1; : : :; Un ) = fF : F [Ui and F \ Ui 6= ;; i = 1; : : :; ng; where each Ui is a nonempty open subset of X contains an element F such that H1(F ) = 1. Since these sets form a basis for the topology of K(X ), H1 has value 1 on a dense set. Thus, H1 cannot have a point of continuity, whereas a Baire class one function has a dense set of points of continuity. This also shows that dimH has value 0 on a dense set and also has value 1 on a dense set and is therefore also not a Baire class 1 function. 3. Packing and box dimension functions
The packing dimension can be de ned either via the upper box counting (i.e., Minkowski) dimension or via the packing measures. In this section we use the rst approach and in the next section the second. We begin with the de nitions and simple measurability properties of the box counting dimensions. Let K 2 K(X ). For > 0 let N (K ) be the smallest number of open balls of radius that are needed to cover K . The upper and lower box counting dimensions dimB K and dimB K of K are de ned by dimB K = lim sup log N (K ) (? log ) !0
= lim sup log N2?j (K ) (j log 2); j !1
and
inf log N (K ) (? log ) dimB K = lim !0 ?j (K ) (j log 2); = lim inf log N 2 j !1
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES
5
where the second equalities are easily seen to hold. Instead of covering with -balls we can also use packings with -balls. Let P (K ) be the largest integer k such that there are points x1 ; : : :; xk 2 K with d(xi; xj ) > for i 6= j . Then one veri es easily (or see [F] or [M]) that dimB K = lim sup P (K )=(? log ) !0 = lim sup P2?j (K )=(j log 2); j !1
and similarly for dimB K . 3.1. Lemma. The functions dimB and dimB : K(X ) ! [0; 1] are of Baire's class 2. Proof. Evidently fK 2 K(X ) : N (K ) < cg is open for c 2 R and > 0, whence N is upper semicontinuous. Thus the functions gj : K 7! log N2?j (K )=(j log 2) and hk = inf jk gj are also upper semicontinuous. Hence dimB = limk!1 hk is of Baire's class 2. For the upper box counting dimension we use the packing function P . It is easy to verify that it is lower semicontinuous. Hence it follows as above that dimB is of Baire's class 2. We de ne the packing dimension dimp A for A X by
dimp A = inf sup dimB Ki : A i
1 [
i=1
Ki ; Ki 2 K(X ) :
Then the packing dimension has the countable stability property, which the box counting dimension lacks: dimp
[ 1
i=1
Ai = sup dimp Ai for Ai X: i
This leads to the following lemma. It was also essentially proved by Falconer and Howroyd in [FH]. 3.2. Lemma. Suppose d 2 R , K 2 K(X ) and dimp K > d. Then there is a non-empty compact set M K such that dimp (M \ V ) d for all open sets V with M \ V 6= ;. Proof. Let M0 = K . De ne a trans nite sequence of compact subsets of K by recursion as follows. For each ordinal , let M +1 = fx 2 M : dimp (M \ V ) d; for all neighborhoods V of xg. For each limit ordinal , let M = \ d: This is certainly true for = 0. Suppose this claim holds for all < . If = + 1, then since M = M +1 [ (M n M +1); dimp M +1 = dimp M > d: If is a countable limit ordinal, then since M0 = [ < (M n M +1 ) [ M ; again we have dimp M > d. Set M = M . Then M 6= ;: If there were some open set V such that dimp (M \ V ) < d and M \ V 6= ;, then M +1 6= M , a contradiction.
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PERTTI MATTILA AND R. DANIEL MAULDIN1
3.3. Lemma. Let c 2 R and K 2 K(X ). Then dimp K c if and only if for every d < c there is a non-empty compact set M K such that dimB (M \ V ) d for all open sets V with M \ V = 6 ;.
Proof. The \only if" part follows immediately from Lemma 3.2. To verify the \if" part suppose that the stated condition holds and dimp K < c, contrary to what is asserted. Choose d < c for which dimp K < d. Then there are compact sets S1 K1; K2; : : : such that K i=1 Ki and dimB Ki < d for all i. Let M K be non-empty, compact and Ssuch that dimB (M \ V ) d; for all open sets V with M \ V 6= ;. Since M = 1 ( M i=1 \ Ki ); the Baire category theorem implies that M \ Ki has non-empty interior relative to M , for some i . Thus there is an open set V with ; 6= M \ V M \ V M \ Ki , whence dimB (M \ V ) dimB (M \ Ki ) < d; which is a contradiction and completes the proof of the lemma. 3.4. Theorem. For c 2 R the set A = fK 2 K(X ) : dimp K cg is analytic. In particular, the function dimp : K(X ) ! [0; 1] is measurable with respect to B(A). Proof. Let fV1 ; V2; : : : g be a basis for the topology of X . For m, n 2 N de ne Bm;n = f(K; M ) 2 K(X ) K(X ) : M K and either M \ Vn = ; or dimB (M \ Vn ) c ? 1=mg: Then by Lemma 3.3, \ 1 1 \ A = proj1 Bm;n : m=1
n=1
Thus it suces to show that each Bm;n is a Borel set. The sets f(K; M ) : M K g and fM : M \ Vn = ;g are closed. The function M 7! N (M \ Vn ) is easily seen to be upper semicontinuous for all > 0, whence M 7! dimB (M \ Vn ) is Borel measurable. Thus every Bm;n is a Borel set. 4. Packing measure function Let A X and > 0. We say that f(xi; ri )gni=1 is a -packing of A if xi 2 A, 2ri > 0 and ri + rj < d(xi ; xj ) for i, j = 1; : : :; n, i 6= j . Then the closed balls B (xi; ri) are disjoint. We rst de ne the prepacking measures Pg; and Pg by X n n g(2ri) : f(xi; ri )gi=1 is a -packing of A ; Pg; (A) = sup i=1 Pg (A) = lim P (A): !0 g;
If g is continuous we can replace the condition ri + rj < d(xi ; xj ) by ri + rj d(xi; xj ) without changing the de nitions of Pg; and Pg . The following simple lemma is given only for completeness; it will not be needed later on.
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES
7
4.1. Lemma. The function Pg; : K(X ) ! [0; 1] is lower semicontinuous and the function Pg : K(X ) ! [0; 1] is of Baire's class 2. Proof. For c 2 R, 1 [ fK 2 K(X ) : Pg; (K ) > cg = Gn n=1
n where Pn K 2 Gn if and only if there exists a -packing f(xi ; ri )gi=1 of K such that i=1 g (2ri ) > c. Each Gn is open, consequently Pg; is lower semicontinuous. Hence Pg = limn!1 Pg;1=n is of Baire's class 2. Since Pg is not countably subadditive one needs a standard modi cation to get an outer measure out of it. Thus we de ne the packing g-measure for A X by
Pg (A) = inf
X 1
i=1
Pg (Ai ) : A
1 [ i=1
Ai :
If A is compact, the sets Ai can also be taken compact. Then Pg is a Borel regular outer measure. When g(r) = rs we denote Pg = P s . The packing dimension, which was introduced in the previous section, can also be de ned in terms of the packing measures: dimp A = inf fs : P s (A) = 0g: For these relations, see e.g., [TT1], [F] or [M]. The proofs there are in R n but they generalize without changes. Let us indicate why dealing with packing measure is somewhat delicate. This can be seen by carrying out a straightforward logical analysis of its de nition. Let
E = fK 2 K(X ) : Pg (K ) < cg: Then
E = proj1
(K; K1; K2; : : : ) 2 K(X )1 : K
= proj1 (C \ D); where and
1 [ i=1
Ki and
1 X i=1
Pg (Ki ) < c
1 [ 1 C = (K; K1; K2; : : : ) 2 K(X ) : K Ki i=1
1 X 1 D = (K; K1; K2; : : : ) 2 K(X ) : Pg (Ki ) < c : i=1
It follows from Lemma 4.1 that D is a Borel set. But, from its de nition, the set C is a coanalytic set. Also, it is not a Borel set. (If it were, then consider C \ F , where F is the Borel set of all (K; K1; K2; : : : ) such that each Ki is a singleton. The projection of this set on its rst coordinate would be an analytic set. But this is the set of all countable closed sets, a classic example proven by Hurewicz not
8
PERTTI MATTILA AND R. DANIEL MAULDIN1
to bePanalytic, see [H]). Thus, this analysis only yields that E is a so-called PCA (or 12) set and whether these types of sets are measurable is independent of the Zermelo{Fraenkel axioms of set theory, see [J, pp. 528 and 563]. In the following Pr(X ) stands for the set of all Borel probability measures on X . We equip Pr(X ) with the topology of weak convergence. Then it is a complete separable metrizable space. The support of a measure 2 Pr(X ) is denoted by spt . 4.2. Theorem. Suppose there is k < 1 such that g(2r) kg(r) for r > 0. Then for all c 2 R the set fK 2 K(X ) : Pg (K ) cg is analytic. In particular, the function Pg : K(X ) ! [0; 1] is measurable with respect to B(A). Proof. We may assume c > 0. We shall make use of the theorem of Joyce and Preiss [JP] according to which for any compact K with Pg (K ) c there exists a compact M K such that c Pg (M ) < 1. De ning 2 Pr(X ) by (B ) = Pg (M \ B )=Pg (M ) we have spt K and c(L) Pg (L) for all L 2 K(X ). The converse of this holds trivially: if there exists 2 Pr(X ) such that spt K and c(L) Pg (L) for all L 2 K(X ), then Pg (K ) c. It follows that fK 2 K(X ) : Pg (K ) cg = proj1 A where A = f(K; ) 2 K(X ) Pr(X ) : spt K; c(L) Pg (L) for L 2 K(X )g: Thus it suces to show that A is a Borel set. We have
A=S\ where and
1 \
m=1
Am
S = f(K; ) 2 K(X ) Pr(X ) : spt K g
Am = f(K; ) 2 K(X ) Pr(X ) : c(L) Pg;1=m(L) for compact sets L K g: First, the set S is clearly closed. Secondly, (K; ) 2 Am if and only if for every j 2 N and every compact set L K there is a (1=m)-packing f(xi ; ri)gni=1 of L such that n X (c ? 1=j ) (L) < g(2ri): i=1
For a xed j the set of all such pairs (K; ) 2 K(X ) Pr(X ) is open. To see this note that its complement consists of those (K; ) for which there exists a compact set L K such that for every (1=m)-packing f(xi ; ri)gni=1 of L we have (c ? 1=j ) (L)
n X i=1
g(2ri);
and this set is easily seen to be closed. Thus each Am is a G -set, and so is A.
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES
9
4.3. Question. Is Pg B(A)-measurable without the doubling condition g(2r) kg(r)? In the proof we needed the doubling condition since Joyce and Preiss proved their result using it. 5. Diameter-based packing measure and dimension
The de nitions for the upper box counting dimension and packing dimension of Section 3 can also be given in terms of the diameters of balls instead of their radii. More precisely, for K 2 K(X ) let Ne (K ) be the smallest number of open balls of diameter at most that are needed to cover K . If we replace N by Ne in the de nitions of dimB and dimp , we get the same dimensions. For packing measures and the dimensions induced by them the situation is di erent in general metric spaces, and we look at that in some detail. We restrict here to the gauge functions g(t) = ts . Let A X and > 0. De ne
Pes (A) =
sup
n X i=1
d(Bi)s
where the supremum is taken over all nite sequences B1; : : :; Bn of disjoint open balls centered at A and of diameter at most . As before, we then de ne
Pes (A) = lim Pes (A); !0 es (A) =
P and also
inf
X 1
i=1
Pes(Ai )
:A
1 [ i=1
Ai ;
g p A = inf fs : P e s (A) = 0g: dim
g p dimp , and it is easy to give examples of metric spaces, Clearly, Pes P s and dim even compact subspaces of R , where both of these inequalities are strict. We would now like to modify the method of Section 3 to prove the measurability g p . For that we need a formula giving dim g p in terms of a box counting-type of dim dimension. For K 2 K(X ) and j = 1; 2; : : : , let pj (K ) be the largest number n of disjoint open balls B1 ; : : :; Bn centered at K with 2?j?1 < d(Bi) 2?j . Of course, pj (K ) may be zero for some j 's. Set g B K = lim sup log pj (K ) dim j log 2
j !1
g B is of Baire's class (with log 0 = ?1). Then pj is lower semicontinuous and so dim 2.
10
PERTTI MATTILA AND R. DANIEL MAULDIN1
5.1. Lemma. For K 2 K(X ), g pK dim
= inf
g B Ki sup dim
i
:K
1 [ i=1
Ki ; Ki 2 K(X ) :
Only very minor modi cations are needed to carry out the argument of Tricot from [T] for the radius packing measures and dimensions, it is given also in [F] and [M]. We leave the details to the reader. Using Lemma 5.1 one can now argue as before to prove the following theorem. g p : K(X ) ! [0; 1] is B (A)-measurable. 5.2. Theorem. The function dim 5.3. Question. Is Pes B(A)-measurable? The problem here is that in Section 4 we used the theorem of Joyce and Preiss [JP] which does not hold for the measures Pes ; a counterexample has been constructed by Joyce [Jo]. 6. Measurability of sections
We shall now study the measurability of the Hausdor and packing measures and dimensions of sections in a product space. Let T be a Polish space. If E T X , and t 2 T , then let Et = fx 2 X : (t; x) 2 E g. 6.1. Theorem. Let B T X be a Borel set. If Bt is -compact for all t 2 T , then the functions t 7! Hg (Bt) and t 7! dimH Bt , t 2 T , are Borel measurable. Proof. Assume rst that each Bt is compact. Then the map t 7! Bt is Borel measurable (even upper semicontinuous, see [K2, p. 58]) from T into K(X ). Thus, the assertions follow from Theorem 2.1. If each Bt is -compact we use aS result of Saint Raymond [S] to nd Borel sets B1 B2 : : : such that B = 1 n=1 Bn and (Bn )t is compact for all t 2 T , n = 1; 2; : : : . As Hg (Bt ) = limn!1 Hg ((Bn)t ) and dimH Bt = limn!1 dimH (Bn )t, the theorem follows. 6.2. Remark. Dellacherie proved in [D] that t 7! Hs (Bt), and hence also t 7! dimH Bt, is B(A)-measurable provided B is an analytic subset of T X and T and X are compact metric spaces; his proof easily extends to the case where T and X are complete and separable. We shall now show that these functions need not be Borel measurable even when B is a Borel subset of [0; 1] [0; 1]. 6.3. Example. There exists a Borel set B [0; 1] [0; 1] such that the function t 7! dimH Bt, t 2 [0; 1], is not Borel measurable. Proof. Let E = fK : 0 1g be a copy of the Baire space (that is, a space homeomorphic to the irrationals) in K([0; 1]) consisting of pairwise disjoint elements and such that dimH K > 0 for all 2 [0; 1]. To see that such a set E exists, consider f , a typical continuous function with respect to the Wiener measure. For almost all y in the range of f , the level set f ?1fyg has Hausdor dimension 1=2, see [T]. The map y 7! f ?1fyg is one-to-one and Borel measurable. Choose a copy of the Baire space D [0; 1] on which this map is a homeomorphism, see [K1]. Then we can take as E the image of D.
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES 11
Let C be a coanalytic subset of [0; 1] which is not a Borel set. By [K1] there exists a continuous map ' of E onto A = [0; 1] n C . De ne
B = f(x; y) 2 [0; 1] [0; 1] : x = '(K ) and y 2 K for some K 2 E g: Then B is a Borel set. To see this let S = f(x; y; K ) : x = '(K ) and y 2 K g. Then S is a Borel set and the projection map onto the rst two coordinates is one-to-one and maps S onto B whence S is a Borel set by [K1, p. 487]. Also the projection of B into the x-axis is A and for x 2 A, Bx contains some K 2 E , whence dimH Bx > 0. If x 2 C , then Bx = ; and dimH Bx = 0. Thus for the map g, g(x) = dimH Bx , the set g?1f0g = C is not a Borel set and so g is not a Borel function. We now turn to the measurability of the packing dimension and measure of the sections. Applying Theorems 3.4, 5.2 and 4.2 we can use exactly the same argument as in 6.1 to prove the following theorem. 6.4. Theorem. Let B T X be a Borel set such that all the sections Bt , t 2 T , are compact. g p Bt , t 2 T , are B (A)-measurable. (1) The functions t 7! dimp Bt and t 7! dim (2) If g is as in Theorem 4.2 the function t 7! Pg (Bt) is B(A)-measurable. 7. An example
In this section we show that the packing dimension and measure functions need not be Borel measurable. For this we shall use continued fractions, see e.g., [R]. Any z 2 [0; 1] can be written as
z = [b1; b2; : : : ] =
1
b1 + b +1 : : : 2
where bi 2 N . The sequence b1 ; b2; : : : is nite if and only if z is rational. We then write [b1; : : :; bj ] = pj =qj where the integers pj and qj are as in [R, p. 136]. The following basic relations hold for them, see (3) and (5) on page 136 of [R]: de ning p?1 = 1, q?1 = 0, p0 = b0, q0 = 1, we have (7.1) (7.2)
pj = bj pj?1 + pj?2 ; qj = bj qj?1 + qj?2 for j = 1; 2; : : :; pj qj?1 ? pj?1 qj = (?1)j?1 for j = 1; 2; : : ::
For irrationals z = [b1; b2; : : : ] we use the fact that the even convergents [b1; b2; : : :; b2j ] increase to z and the odd convergents [b1; b2; : : :; b2j+1 ] decrease to z. Also, [b1; : : :; bj ; b] is increasing with b if j is odd and decreasing if j is even. For the denominators qj we shall need the following simple estimates.
12
PERTTI MATTILA AND R. DANIEL MAULDIN1
7.3. Lemma. b1 : : : bj qj 2j b1 : : : bj for j = 1; 2; : : : .
Proof. Since q1 = b1 and q2 = b1b2 +1, this holds for j = 1; 2. Suppose the asserted inequalities are valid for j < m. Then by (7.1) b1 : : : bm bm (b1 : : : bm?1 ) + qm?2 bm qm?1 + qm?2 = qm bm (2m?1b1 : : : bm?1) + 2m?2b1 : : : bm?2 2m b1 : : : bm: Thus the lemma follows by induction. We denote by I = [0; 1] n Q the set of irrationals in [0; 1]. Let z = [a1; a2; : : : ] 2 I , and de ne N (z) = f[a1; k1; a2; k2; : : : ] : ki 2 N for i = 1; 2; : : : g: 7.4. Lemma. For z 2 I , dimp N (z) 1=2. Proof. We shall show that N (z) contains a compact set M such that dimB (M \ V ) 1=2 whenever V is an open set with M \ V 6= ;. This implies dimp N (z) 1=2 by Lemma 3.3. Temporarily x n1 ; n2; : : : 2 N , ni 11. Let M = f[a1; k1; a2; k2; : : : ] : 1 ki ni ; ki 2 N ; for i = 1; 2; : : : g: Then M is a compact subset of N (z). Let J = J (a1; k1; : : :; am ; km) be the smallest interval containing all the points of M whose expansions begin with a1; k1; : : :; am ; km. Using the above mentioned monotonicity properties of [b1; : : :; bj ; b], we see that J contains the interval with left endpoint A = [a1; k1; : : : ; am ; km; am+1; 2] and right endpoint B = [a1 ; k1; : : :; am; km; am+1; nm+1]. Let pi and qi , i = 0; 1; : : :; 2m + 1, be the partial numerators and denominators generated by the sequence a1; k1; : : :; am; km; am+1 with p0 = 0, q0 = 1. Then by (7.1) A = u=v where u = 2p2m+1 + p2m and v = 2q2m+1 + q2m. Similarly, B = U=V where U = nm+1p2m+1 + p2m and V = nm+1q2m+1 + q2m. Using (7.2) we estimate the length of the interval [A; B ]: 2m ? p2m q2m+1 ) B ? A = ((nnm+1 q? 2)(p+2mq+1q)(2 m+1 2m+1 2m q2m+1 + q2m ) nm+1 ? 2 = (n q m+1 2m+1 + q2m )(2q2m+1 + q2m ) > (n nm++11)?3q22 4q21 ; m+1 2m+1 2m+1 as nm+1 11. Hence, recalling Lemma 7.3, the length of J satis es H1 ?J (a1; k1; : : :; am; km) 4q21 2m+1 42m+2(a : : : a 1 )2(n : : : n )2 1 m+1 1 m := m :
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES 13
We want to show now that two di erent intervals as above are disjoint. By the basic monotonicity properties stated at the beginning of this section, it suces to consider J1 = J (a1; k1; : : :; km?1; am; k) and J2 = J (a1; k1; : : :; km?1; am; k0 ) where 1 k < k0 nm . These two sequences generate the same pi and qi for i < 2m. The interval J1 lies to the left of the interval J2 . The right hand endpoint of J1 is at most B1 = [a1; k1; : : :; am; k; am+1; nm+1 + 1]: So B1 = U1 =V1 where by (7.1) ?
U1 = (nm+1 + 1) (kam+1 + 1) + k) p2m?1 + am+1p2m?2 + kp2m?1 + p2m?2 ? ? = (nm+1 + 1)(kam+1) + k p2m?1 + (nm+1 + 1) am+1 + 1 p2m?2 = p2m?1 + p2m?2 ; and similarly, with the same and ,
V1 = q2m?1 + q2m?2 : The left hand endpoint of J2 is at least
A2 = [a1; k1; : : :; am; k0; am+1; 1] = u2 =v2 where, as above
u2 = 0 p2m?1 + 0 p2m?2 ; v2 = 0 q2m?1 + 0 q2m?2 with Thus by (7.2)
0 = k0 am+1 + 1 + k0 ; 0 = am+1 + 1: A2 ? B1 = u2V1v ?V U1 v2 2 1 0 0 )(p2m?1 q2m?2 ? p2m?2 q2m?1 ) ( ? = v2 V1 0 0 = v ?V : 2 1
Now ?
0 ? 0 = (k0 am+1 + k0 + 1) (nm+1 + 1) am+1 + 1 ? ? (nm+1 + 1)(kam+1 + 1) + k (am+1 + 1) ? = (nm+1 + 1) a2m+1 + 1 (k0 ? k) + (nm+1 + 2) am+1(k0 ? k) ? nm+1 > 0; whence A2 ? B1 > 0 so that J1 and J2 are indeed disjoint.
14
PERTTI MATTILA AND R. DANIEL MAULDIN1
It follows now that at least n1 : : : nm intervals of length m are needed to cover M . Thus m (M ) dimB M lim sup log?Nlog m m!1 n1 : : : n m ) lim sup (2m + 2) log 4 + 2 log( log(a1 : : : am) + 2 log(n1 : : : nm) : m!1
Choosing the sequence (nm ) to grow suciently fast, the last upper limit is 1=2 and then dimB M 1=2. The same argument shows that also dimB (M \ V ) 1=2 for all open sets V with M \ V 6= ;. Thus dimp M 1=2 as required. For A R 2 let Ax = fy 2 R : (x; y) 2 Ag: 7.5. Theorem. a) There exists M 2 K(R 2 ) such that fx 2 R : dimp (Mx) > 0g is an analytic non-Borel set. b) The set E = fK 2 K([0 ; 1]) : dimp K > 0g is an analytic non-Borel set. P1 Indeed, E is a complete 1 set: if A is an analytic subset of a Polish space X , then there is a Borel measurable map h of X into K([0; 1]) such that h?1 (E ) = A. (Complete sets are discussed in [KL].) Proof. The results in Sections 3 and 4 show that the sets in question are analytic. For the claim that they are not Borel sets a) clearly implies b), since x 7! Mx is P Borel measurable. To prove the theorem it suces to prove E is 11 complete. To do this we shall make some additional observations to a beautiful technique of Mazurkiewicz and Sierpinski, [MS]. Let A be an analytic subset of a Polish space X . Let f be a continuous map of I = N N onto A. For z = (a1; a2; : : : ) 2 I , let
'(z) = (a1; a3; a5; : : : ): Then ' is a continuous map of I onto itself. Let g = f ' : I ! A, which is also continuous and onto. Moreover
g?1fxg N (a1; a2; : : : ) whenever x 2 A and f (a1; a2; : : : ) = x. Thus by Lemma 5.1 dimp g?1fxg 1=2 for x 2 A: Let M be the closure of f(g(); ) : 2 Ig. Then M is a closed subset of X [0; 1]. If x 2 A, then Mx g?1fxg, whence dimp Mx 1=2. If x 2 [0; 1] and y 2 Mx, then there is a sequence i 2 I such that (g(i); i) ! (x; y). If y 2 I , then x = lim g(i) = g(y) 2 A. Thus for x 2 [0; 1] n A, Mx Q and so dimp Mx = 0. Hence we have shown that h?1 (E ) = A, where h(x) = Mx .
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES 15
8. Densities
In this section we shall study the lower density ? P A \ B (x; r) g dg (A; x) = lim inf r!0 g(2r) of packing measures Pg for x 2 X and A X . We shall always assume that the gauge function g satis es the doubling condition (8.1) g(2r) kg(r) for r > 0 with some k < 1. We also denote ds (A; x) = dg (A; x) when g(r) = rs : If A R n and Pg (A) < 1, then dg (A; x) = 1 for P s almost all x 2 A according to [ST, Corollary 7.2]. We shall show in Example 8.9 that this is false even for g(r) = rs in any in nite-dimensional Hilbert space. However, in Theorem 8.3 we derive the optimal inequalities 2?s ds (A; x) 1. For this we need a modi cation of well-known covering lemmas. 8.2. Lemma. Let 0 < < 1=2, let B be a collection of closed balls in X such that supfd(B ) : B 2 Bg < 1; and suppose that A X is such that every x 2 A is a center of some B 2 B. Then there exist B (xi ; ri) 2 B, i = 1; 2; : : : , such that the balls B (xi ; ri) are disjoint and [ A B (xi ; ri): i
?
Proof. For each x 2 A choose some B (x) = B x; r(x) 2 B. Set M = supfr(x) : x 2 Ag: Let 2 < t < 1 and de ne A1 = fx 2 A : tM < r(x) M g: Choose a subset B1 A1 which is maximal with respect to the property: if x; y 2 B1 and x 6= y, then x 2= B (y) or y 2= B (x). ? S Then A1 x2B1 B (x) and the balls B x; r(x) , x 2 B1, are disjoint. Next let
A2 = x 2 A n
[
B (x) : t2 M < r(x) tM
x2B1 and choose a maximal subset B2 A2 such that for any
x 2= B (y) or y 2= B (x). It follows that A1 [ A2 ?
[
x2B1 [B2
x; y 2 B2 with x 6= y,
B (x)
and the balls B x; r(x) , x 2 B1 [ B2, are disjoint. Continuing in this manner we nd the desired balls.
16
PERTTI MATTILA AND R. DANIEL MAULDIN1
8.3. Theorem. Let A X and P s (A) < 1. Then 2?s ds (A; x) 1 for P s almost all x 2 A: Proof. The right hand inequality follows with essentially the same proof which was used in [M, 6.10], or one can consult Cutler [Cu] who proved this for Pg with general gauge functions. To prove the left hand inequality, we may assume that A is a Borel set by the Borel regularity of P s . Given > 2 it is enough to show that ds(A; x) ?s for P s almost all x 2 A. Let t < ?s and let
B fx 2 A : ds (A; x) < tg At: Given > 0 we can apply Lemma 8.2 to nd disjoint balls B (xi ; ri=) such that xi 2 B , 2ri < , ?
P s A \ B (xi; ri ) < t(2ri )s and B
1 [
i=1
B (xi; ri ):
Then
1 ? X s P (B ) P s B \ B (xi; ri ) i=1 1 1 X X s s t (2ri) = t (2ri=)s tsPs (B ): i=1 i=1 Letting ! 0 we have P s (B ) ts P s(B ) for all B At , which implies P s (At ) tsP s (At ). As ts < 1, we obtain P s (At ) = 0 for all t < ?s , and this yields that ds(A; x) ?s for P s almost all x 2 A as desired.
8.4. Remark. The inequalities c dg (A; x) 1 for P s almost all x 2 A hold for any g satisfying (8.1) if Pg (A) < 1. Here c is a positive constant (not necessarily optimal) depending on g. This was proved by Cutler in [Cu]. The above proof also applies. Next we show that instead of inequalities we have equality Hg almost everywhere in A. Recall that we are assuming (8.1). 8.5. Theorem. If A X with Pg (A) < 1, then
dg (A; x) = 1 for Hg almost all x 2 A. Proof. By the Borel regularity of Pg we may assume that A is a Borel set. If our claim is false there are t < 1 and a Borel set C A such that Hg (C ) > 0 and
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES 17
dg (A; x) < t for x 2 C (recall Remark 8.4). As Hg Pg (see [Cu, 2.6], also the proof of this fact in R n for g(r) = rs in [M, 5.12] generalizes without diculty), we can use the Radon{Nikodym theorem to nd a non-negative Borel function f on C such that Z Hg (B ) = f dPg B
for Borel sets B C . Since Hg (C ) > 0, there are u > 0 and a Borel set D C such that Hg (D) > 0 and f u on D which yields
Hg (B ) uPg (B ) for B D.
(8.2)
Let > 0. By the analogue of Theorem 8.3 for g(r) = rs, recall Remark 8.4, c < dg (D; x) < t for P s almost all x 2 D. For the upper density with respect to the Hausdor measure we have, cf. [Fe, 2.10.18 (3)], also the proof of [M, 6.2] for g(r) = rs generalizes easily, ?
lim sup Hg D \ B (x; r) =g(2r) 1
(8.3)
r!0
for Hg almost all x 2 X . From (8.2) we see that (8.3) also holds for Pg almost all x 2 D. Let B D be a Borel set. By a standard covering theorem, see e.g., [Fe, 2.8.6], we can nd xi 2 B and 0 < ri < =2 such that the balls Bi = B (xi ; ri) are disjoint,
B
1 [
n [
B (xi; ri) [ B (xi; 5ri ) for all n = 1; 2; : : :; i=1 i=n+1 cg(2ri) Pg (D \ Bi ) tg(2ri) and ? Hg D \ B (xi; 5ri) 2g(10ri):
Thus recalling also (8.1) and (8.2) we have
Pg (B )
n X
Pg (D \ Bi ) +
i=1 n X
t
g(2ri) + u?1
1 X i=n+1 1 X
?
?
Pg D \ B (xi ; 5ri)
Hg D \ B (xi ; 5ri)
i=n+1 1 X 3 ? 1 g(2ri) tPg; (B ) + 2k u i=n+1 1 X 3 ? 1 ? 1 Pg (D \ Bi ): tPg; (B ) + 2k u c i=n+1
Here
i=1
1 X i=n+1
Pg (D \ Bi ) Pg (A) < 1:
18
PERTTI MATTILA AND R. DANIEL MAULDIN1
Thus letting rst n ! 1 and then ! 0, we obtain Pg (B ) tPg (B ) for all Borel sets B D. This implies 0 < Hg (D) Pg (D) tPg (D) < 1 which is a contradiction since t < 1. 8.6. Corollary. Let g satisfy the doubling condition. Let A X with Pg (A) < 1. Then Hg (A) = Pg (A) if and only if ? H A \ B (x; r) g lim = 1 for Pg almost all x 2 A. r!0 g(2r) The proof of Theorem 6.12 of [M] applies without any essential changes when we have Theorem 8.5 at our disposal. 8.7. Remark. An interesting application of Corollary 8.6 in R n is the following theorem of Saint Raymond and Tricot. Let A R n with 0 < P s (A) < 1. Then Hs(A) = P s (A) if and only if s is an integer and A is P s -recti able in the sense that P s almost all of A can be covered with countably many Lipschitz images of R s , see [ST] or [M]. Attempts to extend this to more general metric spaces starting from 8.6 boil down to the following: in which metric spaces X is it true that if A X with Hs(A) < 1, then s ?A \ B (x; r) H s almost all x 2 A = 1 for H lim s r!0 (2r) if and only if s is an integer and Hs almost all of A can be covered with countably many Lipschitz images of R s ? Kirchheim proved in [K] that the \if" part is valid in any metric space. Chlebk has given in [C] a proof which shows that the \only if" part is valid at least in all in nite-dimensional inner product spaces. Thus combining these results with Corollary 8.6 the argument to prove Theorem 17.11 in [M] shows that the theorem of Saint{Raymond and Tricot is valid in Hilbert spaces. 8.8. Theorem. Let X be a Hilbert space, s > 0 and A X with 0 < P s (A) < 1. Then Hs (A) = P s (A) if and only if s is an integer and A is P s -recti able. Now we construct an example in an in nite-dimensional space to show that the lower bound in Theorem 8.3 is the best possible. 8.9. Example. Let X be an in nite-dimensional Hilbert space and s > 0. Then there is a compact set K X such that 0 < P s (K ) < 1 and ds(K; x) 2?s for x 2 K: Proof. Let e1 ; e2; : : : be an orthonormal of X . Let (mk ) be a sequence of p basis s positive integers such that m1 > (1 + 2)
(8.4)
mk % 1 and
1 X
k=1
m?k 1 = 1:
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES 19
De ne the positive numbers k by
mk sk = sk?1 with 0 = 1:
(8.5)
Then k =k?1 ! 0. For i = 1; : : :; m1 set
p
x(i) = 1 ei and B (i) = U (xi ; 1= 2) where U (x; r) denotes the open ball with center x and radius r. Suppose then that for some k 2 N the points x(i1; : : :; ik ) have been de ned. We put
x(i1; : : :; ik ; i) = x(i1 ; : : :; ik ) + k+1 ei ; i = 1; : : :; mk+1: Set
p
?
B (i1; : : :; ik ) = B x(i1; : : :; ik ); k = 2 : Q Let = 1 i=1 f1; : : :; mi g have the product topology. The system fB (i1 ; : : : ; ik ) : 1 ij mj ; j k; k 2 N g of open balls is a Cantor scheme: (8.6) for each k the balls B (i1 ; : : :; ik ), 1 ij mj , 1 j k, are pairwise disjoint. for each k and 1 i mk+1 ,
(8.7)
B (i1; : : :; ik ; i) B (i1; : : :; ik ): p To verify these, recall that mk > (1 + 2)s . Moreover, ?
p
d B (i1 ; : : :; ik ; i) = ( 2 k+1 )s
(8.8) and (8.9) De ne Then (8.10)
mX k+1 i=1
K=
? ? d B (i1 ; : : :; ik ; i) s = d B (i1 ; : : :; ik ) s:
1 [ \ k=1 i1 :::ik
B (i1; : : : ; ik ) =
1 [ \ k=1 i1 :::ik
?
B (i1; : : :; ik ):
K \ B (i1; : : :; ik ) B x(i1; : : : ; ik ); 2k+1 :
Let g be the coding map from the coding space onto K . Thus,
fg()g =
1 \ k=1
B (jk)
20
PERTTI MATTILA AND R. DANIEL MAULDIN1
for = (i1 ; i2; : : : ) 2 where jk = (i1 ; : : :; ik ). By properties (8.6) and (8.7), g is a homeomorphism of the Cantor space onto K and g maps the cylinder set [i1; : : :; ik ] onto K \ B (i1 ; : : :; ik ). De ne e on the cylinder sets in by
p
e([i1; : : : ; ik ]) = ( 2 k )s: By properties (8.8) and (8.9), e satis es Kolmogorov's consistency conditions. Let e also denote the extension of e to a Borel measure on . Let be the image measure on K of e via the coding map g. Thus (8.11)
p ? ? K \ B (i1 ; : : :; ik ) = ( 2 k )s = d B (i1; : : :; ik ) s :
We use to show that P s (K ) > 0. Let A K be a Borel set and > 0. Then there is a disjoint subfamily Bj = U (xj ; rj ) of the S family fB (i1; : : :; ik )g such that d(Bj ) < , Bj \ A 6= ;, say xj 2 Bj \ A, and A j Bj . In fact, by choosing for every x 2 B the largest B = B (i1 ; : : :; ik ) such that x 2 B and d(B ) < , we nd the sequence (Bj ). Using (7) and the fact that k+1 =k ! 0 we see that for ? 0 0 some rj and xj with rj 1+ o() rj0 , x0j 2 K we have Bj0 = B (xj ; rj0 ) Bj . Thus by (8.11),
(A)
X ?
j
(Bj ) =
X
j
?
d(Bj )s 1 + o()
X
1 + o() Ps (A):
j
d(Bj0 )s
Hence (A) P s(A) for all Borel sets A K , which gives 0 < (K ) P s (K ) as desired. We now show that P s (K ) < 1 and prove the lower density estimate. Both of these will follow from ?
p
P s K \ B (j`) ( 2 ` )s for 2 , ` = 1; 2; : : :: (In fact, it is easy to show that also the converse inequality in (8.12) holds.) Let 2 , ` 1, 0 < < `+1 and let D1 ; : : :; Dn be disjoint closed balls centered at K \ B (j`) with (8.12)
dj = d(Dj ) < :
(8.13) We want to show that (8.14)
X
j
?
p
dsj 1 + o() ( 2 ` )s
which clearly yields (8.12). For each j let kj be the unique integer such that (8.15)
p
p
2 2 kj +1 < dj 2 2 kj ;
MEASURE AND DIMENSION FUNCTIONS: MEASURABILITY AND DENSITIES 21
and let Bj = B (i1; : : :; ikj ) be such that jkj = (i1 ; : : :; ikj ) and K \ Bj \ Dj 6= ;. Since k+1 =k ! 0 as k ! 1, we may assume, in order to prove (8.14), that
p
p
2 2 kj +1 + 4kj +2 < dj < 2 2 kj ? 4kj +1 : This has the e ect that K \ Bj Dj , whence the balls Bj are disjoint. It follows then from (8.6){(8.9) that X
(8.16)
p
d(Bj )s d B (j`) s = ( 2 ` )s : ?
Set
p
I = fj : dj 2 kj + 4kj +1 g and p J = fj : dj > 2 kj + 4kj +1 g: Then by (8.16) (8.17)
X
j 2I
dsj
X
j 2I
p
( 2 kj + 4kj +1 )s
?
X
?
1 + o()
j 2I
p
p
?
( 2 kj )s = 1 + o()
X
1 + o() ( 2 ` )s :
j 2I
d(Bj )s
For each k every ball B (i1; : : :; ik?1 ) can contain at most one Dj with kj = k and j 2 J because of the disjointness. Thus there are at most m1 ; : : :; mk?1 indices j 2 J with kj = k. Hence by (8.15) and (8.5) X
j 2J;kj =k
So by (8.15) and (8.4), X
(15)
p
p
dsj m1 : : : mk?1 (2 2 k )s = (2 2)s=mk :
j 2J
p
dsj (2 2)s
X
k+1
