Hausdorff measures of different dimensions are not Borel isomorphic

András Máthé^{∗}

September 25, 2006

Abstract

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H^{s}) and (ℝ, B, H^{t}) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H^{d} is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss.

To prove our result, we apply a random construction and show that for every Borel function f : ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that f(C) has Hausdorff dimension ≤ d.

We also prove this statement in a more general form: If A ⊂ ℝ^{n} is Borel and f : A → ℝ^{m} is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d · dim A and dim f(B) ≤ d · dim f(A).

1 Introduction

The question whether Hausdorff measures of different dimensions are Borel isomorphic or not, has been around for several years. This problem is attributed to B. Weiss and D. Preiss, see also [5]. Let H^{d} denote the d-dimensional Hausdorff measure and let B denote the σ-algebra of Borel subsets of ℝ.

Theorem 1.1. For every 0 ≤ d_{1} < d_{2} ≤ 1 the measure spaces (ℝ, B, H^{d_{1}}) and (ℝ, B, H^{d_{2}}) are not isomorphic. Moreover, there does not exist a Borel bijection f : ℝ → ℝ such that for any Borel set B ⊂ ℝ

0 < H^{d_{1}}(B) < ∞ ⟺ 0 < H^{d_{2}}(f(B)) < ∞.  (1)

The analogous theorem in ℝ^{n} holds, too (see Theorem 5.7).

On the other hand, M. Elekes [1] has proved that the continuum hypothesis implies that the measure spaces (ℝ, M_{H^{s}}, H^{s}) and (ℝ, M_{H^{t}}, H^{t}) are isomorphic whenever s, t ∈ (0, 1), where M_{H^{d}} is the σ-algebra of measurable sets with respect to H^{d}.

In the same article, M. Elekes suggests a method to give a partial solution to the Borel isomorphism problem (Theorem 1.1) and asks the following question.

Question 1. Fix 0 < α < 1. Is it true that every Borel function f : ℝ → ℝ is Hlder continuous of exponent α on a set H_{f} of Hausdorff dimension 1 − α?

The author of the present article has answered Question 1 in the positive [2].

It is easy to see that the positive answer implies that t ≤ \frac{s}{1−s} whenever the s-dimensional and the t-dimensional Hausdorff measures are Borel isomorphic. Unfortunately this approach does not seem to lead to Theorem 1.1 in its whole generality. Note that 1 − α is the best we can have for the dimension of H_{f}, since a typical continuous function is not Hlder continuous of exponent α on any set of dimension larger than 1 − α, as shown by M. Elekes in [1].

Let dim H denote the Hausdorff dimension of the set H.

Theorem 1.2. Let f : ℝ → ℝ be Borel (or Lebesgue) measurable. For every 0 ≤ d ≤ 1 there exists a compact set C ⊂ ℝ such that dim C = d and dim f(C) ≤ d.

∗Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.

2000 Mathematics Subject Classification: Primary 28A78; Secondary 60J80, 60D05 Key words: Hausdorff measure, isomorphism, Hausdorff dimension, Borel function, random construction

Theorem 1.2 clearly implies Theorem 1.1: Let f be Borel measurable and choose a d for which d_{1} < d < d_{2}. By applying Theorem 1.2 we get a compact set C of dimension d with dim f(C) ≤ d. Since d_{1} < d, there exists a Borel subset B of C for which 0 < H^{d_{1}}(B) < ∞ (see e.g. [3]). Now f(B) ⊂ f(C), so it has dimension at most d, which implies that H^{d_{2}}(f(B)) = 0. So f cannot be an isomorphism of the measure spaces (ℝ, B, H^{d_{1}}) and (ℝ, B, H^{d_{2}}), and cannot satisfy (1).

To prove Theorem 1.2 it is clearly enough to show the following.

Theorem 1.3. Suppose that K is a compact set of positive Lebesgue measure and f : K → ℝ is continuous. For every 0 ≤ d ≤ 1 there exists a compact set C ⊂ K of Hausdorff dimension d such that f(C) has Hausdorff dimension at most d.

The sketch of the proof is the following. We define a large class of random constructions such that each of them gives a Cantor set F of dimension at most d almost surely (Section 3). Then, for the given K and f, we choose a random construction of this class which gives a set F for which F ⊂ f(K) and dim f^{−1}(F) ≥ d almost surely. This will imply the theorem with a simple additional argument (Section 4).

As it can be expected, Theorem 1.2 has the following generalisation (proved in Section 5).

Theorem 1.4. Let A ⊂ ℝ^{n} be a Borel set and f : A → ℝ^{m} Borel measurable. Then for every 0 ≤ d ≤ 1 there exists a Borel set B ⊂ A such that dim B = d · dim A and dim f(B) ≤ d · dim f(A).

Notation. Let λ denote the one dimensional Lebesgue measure. For a (Borel) measure μ let I_{t}(μ) denote the t-dimensional energy of μ; that is, I_{t}(μ) = ∫∫ |x − y|^{−t} dμ(x) dμ(y). For Borel measures μ_{k} (k ∈ ℕ) and μ, μ_{k} → μ denotes that μ_{k} weakly converges to μ. Let supp μ denote the support of the measure μ.

We denote by ℕ the set of non-negative integers. We identify each natural number with the set of its predecessors: n = {0, 1, …, n − 1}.

By diam H we mean the diameter of the set H. Let H_{∞}^{s} denote the s-dimensional Hausdorff pre-measure; that is, for any H ⊂ ℝ

H_{∞}^{s}(H) = inf { ∑_{n∈ℕ}(diam I_{n})^{s} : {I_{n}}_{n∈ℕ} is a sequence of intervals and H ⊂ ∪_{n∈ℕ}I_{n}}.

2 Preliminaries

We start with some (probably well-known) statements which we shall use in the sequel.

Lemma 2.1. Suppose that μ and μ_{k} (k ∈ ℕ) are probability measures on ℝ such that μ_{k} → μ. Then μ_{k} × μ_{k} → μ × μ.

Proof. We have to show that for every compactly supported continuous function h : ℝ^{2} → ℝ, ∫_{ℝ^{2}} h d(μ_{k} × μ_{k}) → ∫_{ℝ^{2}} h d(μ × μ). Clearly it is enough to show this for a dense subset of the compactly supported continuous functions. It is well known that functions of the form

∑_{i=1}^{n} f_{i}(x)g_{i}(y) (f, g : ℝ → ℝ continuous functions with compact support)

are dense, so it is enough to check that

∫_{ℝ^{2}} f(x)g(y) d(μ_{k} × μ_{k}) → ∫_{ℝ^{2}} f(x)g(y)d(μ × μ).

By Fubini,

∫_{ℝ^{2}} f(x)g(y) d(μ_{k} × μ_{k}) = ∫_{ℝ} f(x) dμ_{k}(x) ∫_{ℝ} g(y) dμ_{k}(y)

which tends to

∫_{ℝ} f(x) dμ(x) ∫_{ℝ} g(y) dμ(y) = ∫_{ℝ} f(x)g(y) d(μ × μ)

as k → ∞, using μ_{k} → μ and Fubini again.

Lemma 2.2. Suppose that μ_{k} (k ∈ ℕ) are probability measures on ℝ with support in [−R, R] for some R > 0. If μ_{k} → μ then I_{t}(μ) ≤ lim inf I_{t}(μ_{k}).

Proof. Let ϕ be a compactly supported continuous function on the plane which equals 1 on the square [−R, R]^{2} and for which 0 ≤ ϕ(x, y) ≤ 1 everywhere. For each positive integer i define h_{i} : ℝ^{2} → ℝ by setting

h_{i}(x, y) = ϕ(x, y) · min(|x − y|^{−t}, i).

Using Lemma 2.1 we have

∫ h_{i}(x, y) dμ dμ = lim_{k} ∫ h_{i}(x, y) dμ_{k} dμ_{k} ≤ lim inf_{k→∞} ∫ |x − y|^{−t} dμ_{k} dμ_{k} = lim inf_{k→∞} I_{t}(μ_{k}).

The support of μ × μ is in [−R, R]^{2} since the support of μ_{k} is in [−R, R] for all k, so we have

lim_{i→∞} ∫ h_{i}(x, y) dμ(x) dμ(y) = ∫ |x − y|^{−t} dμ(x) dμ(y) = I_{t}(μ).

Thus I_{t}(μ) ≤ lim inf_{k→∞} I_{t}(μ_{k}).

Lemma 2.3. Let 0 < t < 1, H be a compact set in ℝ and I = [0, λ(H)] an interval. Then

∫_{H} ∫_{H} |x − y|^{−t} dλ(x) dλ(y) ≤ ∫_{I} ∫_{I} |x − y|^{−t} dλ(x) dλ(y) = c_{t}λ(H)^{2−t}

where c_{t} is a constant depending only on t.

Proof. Let φ : H → [0, λ(H)] be the following function:

φ(h) = λ((−∞, h] ∩ H).

Using first the fact that φ is a contraction and then that it is a measure preserving transformation between λ|_{H} and λ|_{I}, we obtain

∫_{H} ∫_{H} |x − y|^{−t} dλ(x) dλ(y) ≤ ∫_{H} ∫_{H} |φ(x) − φ(y)|^{−t} dλ(x) dλ(y)

= ∫_{I} ∫_{I} |x − y|^{−t} dλ(x) dλ(y) = ∫_{[0,1]} ∫_{[0,1]} |λ(H)x′ − λ(H)y′|^{−t} λ(H)^{2} dλ(x′) dλ(y′)
= λ(H)^{2−t} ∫_{[0,1]} ∫_{[0,1]} |x′ − y′|^{−t} dλ(x′) dλ(y′) = c_{t}λ(H)^{2−t}

where c_{t} is finite if t < 1.

3 Random construction and upper estimate

Let M ≥ 3 and m be integers with 2 ≤ m ≤ M − 1. Let

M^{<ω} = {(i_{0}, i_{1}, …, i_{n−1}) : n ∈ ℕ, i_{j} ∈ {0, 1, …, M − 1} = M}.

We will consider M^{<ω} as a set of multi-indices and also as the M-adic tree with root ∅, where every node has M children. For an i ∈ M^{<ω} let |i| denote the length of the multi-index; that is, the level of the node i. Definition 1. A representation of M^{<ω} is a mapping ϕ which maps each node i to a non-trivial compact interval ϕ(i) ⊂ ℝ such that

• for every node i and its children ij (j ∈ M) we have ϕ(ij) ⊂ ϕ(i), and

• for every two distinct j, j′ ∈ M, ϕ(ij) and ϕ(ij′) can have at most one point in common.

Now we shall choose a “random m-adic subtree” S of M^{<ω} in the following way. Let X_{i} (i ∈ M^{<ω}) be independent random variables with uniform distributions over the set of m-element subsets of M. That is, for each set T ⊂ {0, 1, …, M − 1} of m elements

ℙ(X_{i} = T) = \frac{1}{\binom{M}{m}} .

Define the random subtree as

S = {(i_{0}, i_{1}, …, i_{n−1}) ∈ M^{<ω} : i_{j} ∈ X_{(i_{0},i_{1},...,i_{j−1})} for every 0 ≤ j ≤ n − 1}.

So ∅ ∈ S, and for each i ∈ S exactly m children of i are in S. It is easy to see that

|{i ∈ S : |i| = n}| = m^{n}

for every n ∈ ℕ.

Given a representation ϕ of M^{<ω}, consider the closed sets F_{i} = ϕ(i) (i ∈ M^{<ω}) and the random closed sets

F^{n} = ∪{F_{i} : i ∈ S, |i| = n} (n ∈ ℕ).

Then F_{∅} = F^{0} ⊃ F^{1} ⊃ F^{2} ⊃ · · ·. Put

F = ⋂_{n} F^{n}.

We can consider F as the image of the random m-adic subtree S.

Proposition 3.1. For any representation of M^{<ω}, the random closed set F defined above has Hausdorff dimension at most \frac{log m}{log M} almost surely.

Proof. Let 1 > s > \frac{log m}{log M} be arbitrary and q = \frac{m}{M^{s}}, thus q < 1. We cover F^{n} with those intervals F_{i}, for which |i| = n and i ∈ S. For any i of length n we have

ℙ(i ∈ S) = ( \frac{m}{M} )^{n} ,

hence

𝔼( ∑_{\begin{matrix}|i|=n \\ i∈S \end{matrix}}(diam F_{i})^{s}) = ( \frac{m}{M} )^{n} ∑_{|i|=n}(diam F_{i})^{s}.

Since the intervals F_{i} (|i| = n) are almost disjoint (two of them can only have one point in common), ∑_{|i|=n} diam F_{i} ≤ D =^{def} diam F_{∅}. Thus, applying Jensen’s inequality to the concave function x ↦ x^{s}, we obtain

( \frac{m}{M} )^{n} ∑_{|i|=n}(diam F_{i})^{s} ≤ ( \frac{m}{M} )^{n} M^{n} ( \frac{D}{M^{n}} )^{s} = D^{s} ( \frac{m}{M^{s}} )^{n} = D^{s}q^{n}.

Therefore

𝔼(H_{∞}^{s}(F)) ≤ 𝔼(H_{∞}^{s}(F^{n})) ≤ 𝔼( ∑_{\begin{matrix}|i|=n \\ i∈S \end{matrix}}(diam F_{i})^{s}) ≤ D^{s}q^{n}.

Since this is true for every n, we get that

𝔼(H_{∞}^{s}(F)) = 0,

thus H_{∞}^{s}(F) = 0 almost surely, so H^{s}(F) = 0 almost surely. Because s > \frac{log m}{log M} can be chosen arbitrarily, the dimension of F is at most \frac{log m}{log M} almost surely.

4 Lower estimate

Proof of Theorem 1.3. If there exists an y ∈ f(K) for which f^{−1}(y) is of positive measure, then we can choose a compact set C ⊂ f^{−1}(y) of arbitrary Hausdorff dimension d (0 ≤ d ≤ 1), and clearly f(C) = {y} has Hausdorff dimension at most d. Thus we may assume that for every y ∈ f(K) the set f^{−1}(y) has Lebesgue measure zero. Without loss of generality we may suppose that λ(K) = 1.

Now we define the particular representation of M^{<ω} which is adequate for our needs. All the endpoints of the intervals ϕ(i) (i ∈ M^{<ω}) will be contained in f(K). We define ϕ(∅) to be the smallest interval which contains f(K). If an interval is already defined, then its M subintervals (its children) are chosen such that their preimages (with respect to f) have equal Lebesgue measure: \frac{1}{M} times the Lebesgue measure of the preimage of the interval. Now we give a more precise definition.

Define ψ : f(K) → ℝ as

ψ(x) = λ({z ∈ K : f(z) ≤ x}).

Since the inverse image of any point in f(K) has measure zero, this is a continuous increasing function, and its image is the interval [0, λ(K)].

For an i ∈ M^{<ω} let

y_{1}^{i} = max \{y ∈ f(K) : ψ(y) = ∑_{j=1}^{|i|} \frac{i_{j−1}}{M^{j}} \},

y_{2}^{i} = min \{y ∈ f(K) : ψ(y) = \frac{1}{M^{|i|}} + ∑_{j=1}^{|i|} \frac{i_{j−1}}{M^{j}} \}.

Let F_{i} = ϕ(i) = [y_{1}^{i}, y_{2}^{i}]. It is obvious from the definition that

F_{(i_{0},...,i_{k−1})} ⊃ ⋃_{j=0}^{M−1} F_{(i_{0},...,i_{k−1},j)},

λ(f^{−1}(F_{i})) = λ(\{z ∈ K : f(z) ∈ F_{i}\}) = \frac{1}{M^{|i|}}

and that ϕ is a representation of M^{<ω}.

Now let S be a random m-adic subtree of M^{<ω}, and define the random closed sets

F^{n} = ∪{F_{i} : i ∈ S, |i| = n} (n ∈ ℕ),

F = ⋂_{n∈ℕ}F^{n}

the same way as before. From Proposition 3.1, F has Hausdorff dimension at most \frac{log m}{log M} almost surely. Hence F cannot contain an interval, and since all the intervals ϕ(i) (i ∈ M^{<ω}) have their endpoints in f(K), F ⊂ f(K) almost surely.

Let G_{i} = f^{−1}(F_{i}), G^{n} = f^{−1}(F^{n}) and G = f^{−1}(F) be (random) compact sets in K. Then we also have

G^{n} = ∪{G_{i} : i ∈ S, |i| = n} (n ∈ ℕ)

and

G = ⋂_{n∈ℕ}G^{n}.

We claim that G has Hausdorff dimension at least \frac{log m}{log M} almost surely. The key point in our construction was that λ(G_{i}) = \frac{1}{M^{|i|}}, and we also know that λ(G_{i} ∩ G_{i′}) = 0 provided that i ≠ i′

and |i| = |i′|. Note that λ(G^{k}) = ( \frac{m}{M} )^{k}.

We define random Borel measures μ_{k} on ℝ by

μ_{k}(H) = λ(H ∩ G^{k}) · ( \frac{M}{m} )^{k} ,

or equivalently,

μ_{k} = ( \frac{M}{m} )^{k} · λ|_{G^{k}} (k ∈ ℕ).  (2)

Hence μ_{k} is a probability measure with support G^{k} ⊂ K.

Let 0 < t < \frac{log m}{log M} be fixed. We would like to give an upper bound for the expected value of the t-energy of μ_{k}. To do this at first we need to calculate some basic probability. We know that ℙ(i ∈ S) = ( \frac{m}{M} )^{|i|} for every i ∈ M^{<ω}. How much is ℙ(i ∈ S, i′ ∈ S) if |i| = |i′| = k? Let i ∧ i′ denote the nearest common ancestor of i and i′ in the tree M^{<ω}, and let l = l(i, i′) = |i ∧ i|; that is, l is the largest integer for which i_{0} = i′_{0}, i_{1} = i′_{1}, …, i_{l−1} = i′_{l−1} hold (0 ≤ l ≤ k).

ℙ(i ∈ S, i′ ∈ S) = ℙ((i_{j} ∈ X_{(i_{0},...,i_{j−1})} for every 0 ≤ j ≤ l − 1)
and (i_{l}, i′_{l} ∈ X_{(i_{0},...,i_{l−1})})
and (i_{j} ∈ X_{(i_{0},...,i_{j−1})} for every l + 1 ≤ j ≤ k − 1),
and (i′_{j} ∈ X_{(i′_{0},...,i′_{j−1})} for every l + 1 ≤ j ≤ k − 1)).

The random variables X_{i} are independent, so this probability is

= ( \frac{m}{M} )^{l} \frac{m(m − 1)}{M(M − 1)} ( \frac{m}{M} )^{k−l−1} ( \frac{m}{M} )^{k−l−1} = ( \frac{m}{M} )^{2k−l−1} \frac{m − 1}{M − 1} ≤ ( \frac{m}{M} )^{2k−l}  (3)

provided that l < k, that is, i ≠ i′, but the upper estimate clearly holds in the case i = i′ (l = k) as well.

By (2), for any i of length k we have

μ_{k}|_{G_{i}} = \{ \begin{matrix}( \frac{M}{m} )^{k} · λ|_{G_{i}} & if & i ∈ S \\ 0 & if & i ∉ S. \end{matrix}  (4)

Applying first that supp μ_{k} = G^{k} is contained in ∪_{|i|=k}G_{i}, and then (4) and (3),

𝔼 I_{t}(μ_{k}) = 𝔼 (∫∫ |x − y|^{−t} dμ_{k}(x) dμ_{k}(y)) = 𝔼 ( ∑_{|i|=|i′|=k} ∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} dμ_{k}(x) dμ_{k}(y)) =

= ∑_{|i|=|i′|=k} 𝔼 (∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} dμ_{k}(x) dμ_{k}(y)) =

= ∑_{|i|=|i′|=k} ℙ(i ∈ S, i′ ∈ S) ∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} ( \frac{M}{m} )^{k} ( \frac{M}{m} )^{k} dλ(x) dλ(y) =

≤ ∑_{|i|=|i′|=k} ( \frac{m}{M} )^{2k−l(i,i′)} ( \frac{M}{m} )^{2k} ∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} dλ(x) dλ(y) =

= ∑_{|i|=|i′|=k} ( \frac{M}{m} )^{l(i,i′)} ∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} dλ(x) dλ(y). (5)

We denoted the nearest common ancestor of i and i′ by i ∧ i′, let us also use the brief notation h ≤ i ∧ i′ if h is a common ancestor of i and i′. Starting with (5) and then applying Lemma 2.3,

𝔼 I_{t}(μ_{k}) ≤ ∑_{|i|=|i′|=k} ( \frac{M}{m} )^{l(i,i′)} ∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} dλ(x) dλ(y)

= ∑_{l=0}^{k} ( \frac{M}{m} )^{l} ∑_{\begin{matrix}h \\ |h|=l \end{matrix}} ∑_{\begin{matrix}i,i′ \\ h=i∧i′ \\ |i|=|i′|=k \end{matrix}} ∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} dλ(x) dλ(y)

≤ ∑_{l=0}^{k} ( \frac{M}{m} )^{l} ∑_{\begin{matrix}h \\ |h|=l \end{matrix}} ∑_{\begin{matrix}i,i′ \\ h≤i∧i′ \\ |i|=|i′|=k \end{matrix}} ∫_{G_{i}} ∫_{G_{i′}} |x − y|^{−t} dλ(x) dλ(y)

= ∑_{l=0}^{k} ( \frac{M}{m} )^{l} ∑_{\begin{matrix}h \\ |h|=l \end{matrix}} ∫_{G_{h}} ∫_{G_{h}} |x − y|^{−t} dλ(x) dλ(y)

≤ ∑_{l=0}^{k} ( \frac{M}{m} )^{l} ∑_{\begin{matrix}h \\ |h|=l \end{matrix}} c_{t}λ(G_{h})^{2−t} = ∑_{l=0}^{k} ( \frac{M}{m} )^{l} ∑_{\begin{matrix}h \\ |h|=l \end{matrix}} c_{t} ( \frac{1}{M^{l}} )^{2−t} = ∑_{l=0}^{k} ( \frac{M}{m} )^{l} M^{l}c_{t} ( \frac{1}{M^{l}} )^{2−t}

= ∑_{l=0}^{k} c_{t} ( \frac{M^{t}}{m} )^{l} ≤ ∑_{l=0}^{∞} c_{t} ( \frac{M^{t}}{m} )^{l} =^{def} c(t, M, m),

where c(t, M, m) is finite whenever \frac{M^{t}}{m} < 1, that is, t < \frac{log m}{log M}.

By Fatou’s lemma,

𝔼 lim inf_{k→∞} I_{t}(μ_{k}) ≤ lim inf_{k→∞} 𝔼 I_{t}(μ_{k}) ≤ c(t, M, m),

thus lim inf_{k→∞} I_{t}(μ_{k}) is almost surely finite.

Since the probability measures μ_{k} are supported on the same compact set K, every sequence of them has a weakly convergent subsequence. So we can choose a sequence of integers k_{j} such that

lim_{j→∞} I_{t}(μ_{k_{j}}) = lim inf_{k→∞} I_{t}(μ_{k})

and that μ_{k_{j}} is weakly convergent. Let μ = lim_{j→∞} μ_{k_{j}}.

Since supp μ_{k_{j}} = G^{k_{j}} and G^{0} ⊃ G^{1} ⊃ G^{2} ⊃ ⋯, the weak limit μ is supported on ⋂_{j} G^{k_{j}} = G. Applying Lemma 2.2,

I_{t}(μ) ≤ lim inf_{j→∞} I_{t}(μ_{k_{j}}) = lim inf_{k→∞} I_{t}(μ_{k}),

which is almost surely finite. Therefore the compact set G almost surely carries a measure μ with finite t-energy, for any t < \frac{log m}{log M}. Thus the Hausdorff dimension of the set G is at least \frac{log m}{log M} almost surely.

By Proposition 3.1, almost surely both of the inequalities dim F ≤ \frac{log m}{log M} and dim G ≥ \frac{log m}{log M} hold. Hence there exists a compact set G ⊂ K such that dim G ≥ \frac{log m}{log M} and dim f(G) ≤ \frac{log m}{log M}.

For d = 0 or d = 1 the statement of the theorem is trivial, so let 0 < d < 1 be arbitrary. Let

E = \{ \frac{log m}{log M} : M ≥ 3, 2 ≤ m ≤ M − 1\} ,

this is a countable dense set in (0, 1). We constructed compact sets G_{e} for every e ∈ E such that G_{e} is of dimension at least e and f(G_{e}) is of dimension at most e. Let G = ∪_{e<d}G_{e}. Clearly G is a Borel set of dimension at least d, and f(G) = ∪_{e<d}f(G_{e}) is of dimension at most d. It is well known that G contains compact subsets C_{n} of dimension at least d − \frac{1}{n}, and clearly we can require that C_{n} have diameter at most \frac{1}{n}. Let C be the closure of ∪_{n}C_{n}, then C \ ∪_{n}C_{n} is at most one point. Thus C ⊂ K, dim C = d, and clearly dim f(C) ≤ d for the compact set C, which proves the theorem.

5 Generalisation of Theorem 1.2

In this section we shall prove Theorem 1.4. Our first step is to extend Theorem 1.2 in the following way:

Claim 5.1. Let f : [0, 1] → ℝ be a Borel function. For every 0 ≤ d ≤ 1 there exists a compact set C ⊂ ℝ such that dim C = d and dim f(C) ≤ d · dim f([0, 1]).

This is a strengthening of Theorem 1.2 if the image set of f has dimension smaller than 1. To prove Claim 5.1, it is clearly enough to show the following:

Claim 5.2. Suppose that K ⊂ ℝ is a compact set of positive Lebesgue measure and f : K → ℝ is continuous. For every 0 ≤ d ≤ 1 there exists a compact set C ⊂ K such that dim C = d and dim f(C) ≤ d · dim f(K).

To prove this claim we modify the upper estimate we presented in Section 3.

Definition 2. Let ϕ be a representation of M^{<ω}. The support of this representation is the set

K_{ϕ} = ⋂_{k=0}^{∞} ⋃{ϕ(i) : i ∈ M^{<ω}, |i| = k},

and the dimension of the representation is the dimension of K_{ϕ}.

Recall that if S is a random m-adic subtree of M^{<ω}, then we define the random set

F = ⋂_{k=0}^{∞} ⋃{ϕ(i) : i ∈ S, |i| = k}.

Proposition 5.3. For any representation of M^{<ω} of dimension β, the random closed set F defined above has Hausdorff dimension at most \frac{log m}{log M} · β almost surely.

Proof. The case β = 1 is already proved in Proposition 3.1, so we may assume that β < 1 and thus K_{ϕ} (the support of the representation) is a nowhere dense compact and perfect set. Hence considering any infinite branch in M^{<ω}, the diameter of the corresponding intervals tends to zero.

It is easy to see that for each i ∈ M^{<ω},

ℙ(i ∉ S and ϕ(i) ∩ F ≠ ∅) = 0,

thus

ℙ(ϕ(i) ∩ F ≠ ∅) = ℙ(i ∈ S) = ( \frac{m}{M} )^{|i|} .  (6)

Fix any β < t < 1. Since H^{t}(K_{ϕ}) = 0, for any ε > 0 we can choose a finite collection of disjoint open intervals I covering K_{ϕ} such that ∑_{I∈I}(diam I)^{t} < ε, and that each interval I ∈ I intersects K_{ϕ}.

Fix an I ∈ I temporarily. Consider the longest multi-index i ∈ M^{<ω} for which ϕ(i) ⊃ I ∩ K_{ϕ}. At first let us suppose that i has a child i_{I} for which ϕ(i_{I}) ⊂ I. Set l_{I} = |i|, thus |i_{I}| = l_{I} + 1.

From (6) we obtain

ℙ(I ∩ F ≠ ∅) ≤ ℙ(ϕ(i) ∩ F ≠ ∅) = ( \frac{m}{M} )^{l_{I}} .

Now suppose that i has no child i_{I} for which ϕ(i_{I}) ⊂ I. Then it is easy to check that i has two children i_{1} and i_{2} such that

I ∩ K_{ϕ} ⊂ ϕ(i_{1}) ∪ ϕ(i_{2}) and ϕ(i_{j}) ∩ I ∩ K_{ϕ} ≠ ∅ (j = 1, 2).  (7)

Let i′′_{j} be one of the nearest descendants of i_{j} for which ϕ(i′′_{j}) ⊂ I holds (j = 1, 2). Let i′_{j} be the parent of i′′_{j} (j = 1, 2). It is easy to see that

ϕ(i_{j}) ∩ I ∩ K_{ϕ} = ϕ(i′_{j}) ∩ I ∩ K_{ϕ} (j = 1, 2),  (8)

since otherwise we have ϕ(i′_{j}) ⊂ I or ϕ(h) ⊂ I for a sibling h of i′_{j}, contradicting the choice of i′′_{j}. By (7) and (8) we obtain

I ∩ K_{ϕ} ⊂ ϕ(i′_{1}) ∪ ϕ(i′_{2}).  (9)

Set l_{I} = min(|i′_{1}|, |i′_{2}|), and let i_{I} be i′′_{j} (j = 1 or 2) such that |i_{I}| = l_{I} + 1. From (9) and (6) we obtain that

ℙ(I ∩ F ≠ ∅) ≤ ℙ(ϕ(i′_{1}) ∩ F ≠ ∅ or ϕ(i′_{2}) ∩ F ≠ ∅) ≤ ( \frac{m}{M} )^{|i′_{1}|} + ( \frac{m}{M} )^{|i′_{2}|} ≤ 2 ( \frac{m}{M} )^{l_{I}} .

Thus, for all I ∈ I, we defined l_{I} ∈ ℕ and i_{I} of length l_{I} + 1 such that ∅ ≠ ϕ(i_{I}) ⊂ I and

ℙ(I ∩ F ≠ ∅) ≤ 2 ( \frac{m}{M} )^{l_{I}} .  (10)

Since the intervals I ∈ I are disjoint, the nodes i_{I} form an anti-chain in M^{<ω}; that is, none of them is an ancestor of any other. Thus

∑_{I∈I} \frac{1}{M^{|i_{I} |}} ≤ 1,

hence

∑_{I∈I} \frac{1}{M^{l_{I}}} ≤ M.  (11)

Let s = \frac{log m}{log M} · t, hence s < t and m^{t/s} = M. Now cover the random set F ⊂ K_{ϕ} with those intervals I ∈ I which intersect F. By (10),

𝔼(H_{∞}^{s}(F)) ≤ ∑_{I∈I} ℙ(I ∩ F ≠ ∅) · (diam I)^{s} ≤ ∑_{I∈I} 2 ( \frac{m}{M} )^{l_{I}} · (diam I)^{s}

= 2c ∑_{I∈I} \frac{(m^{l_{I} ·t/s} · (diam I)^{t})^{s/t}}{cM^{l_{I}}} , (12)

where we choose c so that ∑_{I∈I} \frac{1}{cM^{l_{I}}} = 1 holds, hence c ≤ M by (11). Applying Jensen’s inequality to the concave function x ↦ x^{s/t} and using m^{t/s} = M we have

2c ∑_{I∈I} \frac{(m^{l_{I} ·t/s} · (diam I)^{t})^{s/t}}{cM^{l_{I}}} ≤ 2c (∑_{I∈I} \frac{m^{l_{I} ·t/s} · (diam I)^{t}}{c · M^{l_{I}}} )^{s/t} = 2c (∑_{I∈I} \frac{1}{c} · (diam I)^{t})^{s/t}

= 2c^{1−s/t} (∑_{I∈I}(diam I)^{t})^{s/t} ≤ 2M (∑_{I∈I}(diam I)^{t})^{s/t} ≤ 2Mε^{s/t} ≤ 2Mε. (13)

Because ε was arbitrarily small, by (12) and (13) we obtain that 𝔼(H_{∞}^{s}(F)) = 0 for every s > β· \frac{log m}{log M}, since β < t < 1 was arbitrary. This implies that the dimension of F is at most β · \frac{log m}{log M} almost surely.

Proof of Claim 5.2. In the proof of Theorem 1.3 in Section 4 we used a representation ϕ which had its support in f(K). So that proof with Proposition 5.3 (instead of Proposition 3.1) instantly gives a compact set C ⊂ K of Hausdorff dimension d such that f(C) has Hausdorff dimension at most d · dim f(K) (instead of d).

Claim 5.4. Let A ⊂ ℝ be compact, f : A → ℝ Borel, dim A > 0, and 0 < s < dim A. For every 0 ≤ d ≤ 1 there exists a Borel set B ⊂ A such that

dim B ≥ d · s and dim f(B) ≤ d · dim f(A).

Proof. It is well-known (see e.g. [3]) that for every s < dim A there exist a probability measure ν with supp ν ⊂ A and a positive constant c such that for every x, y ∈ A we have

ν([x, y]) ≤ c |x − y|^{s} .  (14)

Let us define the continuous function ψ : A → [0, 1] and the Borel function χ : [0, 1] → A by setting

ψ(x) = ν((−∞, x]),
χ(y) = min{x : ψ(x) = y}.

Thus ψ ∘ χ is the identity of [0, 1]. It is easy to check that (14) implies that for every set H ⊂ [0, 1],

dim χ(H) ≥ s · dim H.  (15)

Apply Claim 5.1 to the Borel function f ∘ χ : [0, 1] → ℝ. We get that for every 0 ≤ d ≤ 1 there exists a compact set C ⊂ [0, 1] such that

dim C = d and dim f(χ(C)) ≤ d · dim f(A).

Put B = χ(C). (This is a Borel set, since B = ψ^{−1}(C) ∩ {x ∈ A : χ(ψ(x)) = x}.) Applying (15),

dim B ≥ d · s and dim f(B) ≤ d · dim f(A),

which proves the claim.

Claim 5.5. Let A ⊂ ℝ be a Borel set and let f : A → ℝ be Borel. For every 0 ≤ d ≤ 1 there exists a Borel set B ⊂ A such that dim B = d · dim A and dim f(B) ≤ d · dim f(A).

Proof. We may suppose that dim A > 0. For every sufficiently large positive integer n choose a compact set A_{n} ⊂ A of dimension ≥ dim A − \frac{1}{n}, and apply Claim 5.4 to A_{n} and s = dim A − \frac{2}{n} > 0. We obtain a Borel set B_{n} ⊂ A_{n} for which

dim B_{n} ≥ d · (dim A − \frac{2}{n}) and dim f(B_{n}) ≤ d · dim f(A_{n}) ≤ d · dim f(A).

Now any Borel subset of ∪_{n}B_{n} of dimension d · dim A is an appropriate choice for B.

Lemma 5.6. For each positive integer n there exists a Borel set B_{n} ⊂ ℝ and a Borel bijection p_{n} : B_{n} → ℝ^{n} such that for every set H ⊂ B_{n} we have

dim p_{n}(H) = n · dim H,

moreover, for every 0 ≤ d ≤ 1 and H ⊂ B_{n},

0 < H^{d}(H) < ∞ ⟺ 0 < H^{d·n}(p_{n}(H)) < ∞.

Proof. For x ∈ ℝ let d_{k}(x) ∈ {0, 1, …, 9} (k ∈ ℤ) denote the digits of x in the decimal number system; that is,

x = ∑_{k∈ℤ}d_{k}(x) · 10^{k},

where d_{k}(x) = 0 if k ≥ k_{0} for some k_{0}, and lim inf_{k→∞} d_{−k}(x) ≠ 9. Let

B_{n} = {x ∈ ℝ : ∀ j ∈ {0, 1, …, n − 1} lim inf_{i→∞} d_{j−ni}(x) ≠ 9},

p_{n}^{j}(x) = ∑_{i∈ℤ}d_{j+ni}(x) · 10^{i} (j ∈ {0, 1, …, n − 1})

and

p_{n}(x) = (p_{n}^{0}(x), p_{n}^{1}(x), …, p_{n}^{n−1}(x)).

Hence p_{n} is a Borel bijection between B_{n} and ℝ^{n}. It is easy to check that p_{n} satisfies all the requirements, see [4, Theorem 49] and its proof for a hint.

Proof of Theorem 1.4. Suppose that A ⊂ ℝ^{n} is a Borel set and f : A → ℝ^{m} is Borel measurable. Let d ∈ [0, 1] be arbitrary. Let p_{n} and p_{m} be as in Lemma 5.6. Applying Claim 5.5 to the Borel set p_{n}^{−1}(A) ⊂ ℝ and Borel mapping

p_{m}^{−1} ∘ f ∘ p_{n}|_{p_{n}^{−1}(A)} : p_{n}^{−1}(A) → p_{m}^{−1}(f(A))

we obtain a Borel set B ⊂ p_{n}^{−1}(A) such that

dim B = d · dim p_{n}^{−1}(A) and dim p_{m}^{−1} ∘ f ∘ p_{n}(B) ≤ d · dim p_{m}^{−1}(f(A)).

Using Lemma 5.6 four times we get that

dim p_{n}(B) = d · dim(A) and dim f(p_{n}(B)) ≤ d · dim f(A)

hold for the Borel set p_{n}(B) ⊂ A.

Let B_{n} denote the σ-algebra of Borel subsets of ℝ^{n}. Lemma 5.6 implies that the generalisation of Theorem 1.1 in ℝ^{n} holds.

Theorem 5.7. For every 0 ≤ d_{1} < d_{2} ≤ n the measure spaces (ℝ^{n}, B_{n}, H^{d_{1}}) and (ℝ^{n}, B_{n}, H^{d_{2}}) are not isomorphic. Moreover, there does not exist a Borel bijection f : ℝ^{n} → ℝ^{n} such that for any Borel set B ⊂ ℝ^{n}

0 < H^{d_{1}}(B) < ∞ ⟺ 0 < H^{d_{2}}(f(B)) < ∞.

References

[1] M. Elekes, Hausdorff measures of different dimensions are isomorphic under the Continuum Hypothesis, Real Anal. Exchange 30 (2004/05), no. 2, 605–616.

[2] A. Máthé, Measurable functions are of bounded variation on a set of dimension 1/2, in preparation.

[3] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.

[4] C. A. Rogers, Hausdorff Measures, Cambridge University Press, 1970.

[5] Selected Problems on M. Csörnyei’s homepage

   http://www.homepages.ucl.ac.uk/∼ucahmcs/probl.ps

Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary

Email address: amathe@cs.elte.hu
http://amathe.web.elte.hu