Characterizations of Euclidean spheres

1.   Introduction and main results

Let $\mathcal{X}$ and $\mathcal{Y}$ be two separable metric spaces and let $\mathcal{M}(\mathcal{X})$ be the family of compactly supported Borel probability measures on $\mathcal{X}$. We say that $\mu\in \mathcal M(\mathcal X)$ satisfies the doubling condition if

for some $a > 1$ (or, equivalently, any $a > 1$), where $\mathsf B(x, r)$ is closed ball with a center $x$ and a radius $r$. We use $\mathcal M_0(\mathcal X)$ to denote the family of compactly supported Borel probability measures on $\mathcal X$ that fulfill the doubling condition ^[1]. To study the multifractal analysis of measures introduced by Mandelbrot in ^[2,3], we must turn back to the study of sets related to the local behavior of such measures, called level sets, defined, for $\beta \in \mathbb R$, as :

where $\text{supp} (\mu)$ is the topologic support of $\mu$, and $\mathsf B(x, r)$ stands for the closed ball with a center $x$ and a radius $r > 0$. Thus, this study is essentially linked to its punctual nature and falls under set theory. However, some geometric sets are essentially known by means of the measures that are supported by them, i.e.,

for a given measure $\nu$ and a given set $A$. Hence, when we consider a set $A$, we focus on the properties of the measure $\nu$ rather than the geometric structure of $A$. The set $A$ is thus partitioned into $\alpha$-level sets $E_\mu(\beta)$. This allows the inclusion of $\mu$ into the computation of the fractal measures and dimensions. Olsen, in ^[1], introduced the multifractal generalizations of the fractal dimensions. This is achieved by constructing the generalization of Hausdorff and the packing measures, denoted $\mathcal H_{\mu}^{q, t}$ and $\mathcal P_{\mu}^{q, t}$ in $\mathbb R^d$, where $d\ge 1$, respectively. Later, in ^[4], the authors introduced a new multifractal formalism that deviates from the classical approach. To achieve this, they constructed two distinct measures known as the lower and upper Hewitt-Stromberg (H-S) measures, denoted, respectively, by ${\mathsf H}_{\mu}^{q, t}$ and ${\mathsf P}_{\mu}^{q, t}$. These measures serve as fundamental tools in the analysis of multifractal structures. Given the importance of these measures in this study, it is crucial to examine their properties, including their behavior on product sets and their density characteristics both of which play a critical role in understanding the broader implications of this new formalism. In particular, in ^[5], the author proved the existence of a constant $c > 0$ such that, for any measurable sets $A \subseteq \mathbb{R}^d$ and $B \subseteq \mathbb{R}^l$, the following inequality holds:

provided that we have the measures $\mu\in \mathcal M_0(\mathbb R^d)$ and $\nu \in \mathcal M_0(\mathbb R^l)$ and with the convention that

The constant $c_i$ ($i = 1, 2, 3$) depends only on certain structural parameters, such as the dimensions $d$ and $l$, but is independent of the specific choice of $E$ and $F$. Moreover, in the specific case $q = 0$, the associated dimensional inequalities for the products of these measures have been derived in ^[6,7,8]. For additional related discussions, the readers may consult ^[9,10]. Furthermore, the inequalities above are explicitly stated in this case in ^[8,11,12,13]. In particular, if

are basically the Lebesgue measure on $\mathbb R$, one has, for

and $E = F$ as the middle third Cantor ^[14,15]

Remark 1. The equation of (1.1) has important physical interpretations depending on the context. Note, for $q = 0$, that

is the one-dimensional Lebesgue measure. In particular, if $A, B \subseteq \mathbb{R}$, then the product set $A \times B$ forms a subset of $\mathbb{R}^2$, and then (1.1) gives an approximation of the Hausdorff measure of $A\times B$ using the area of the region covered by the Cartesian product. These prove, in particular, that

A Hausdorff function

is a function that is increasing, continuous, and satisfies

These functions are often used in the context of geometric measure theory, particularly in defining Hausdorff measures. Let $\mathcal{F}$ denote the set of all such dimension functions, i.e., the set of all Hausdorff functions. Additionally, a Hausdorff function $h$ is considered to fulfill the doubling condition if a positive constant $\gamma$ exists such that the following inequality holds:

This condition essentially ensures that $h$ does not grow too quickly and is often used to ensure specific regularity properties of the corresponding measures. The subset of $\mathcal{F}$ consisting of all Hausdorff functions that satisfy the doubling condition is denoted by $\mathcal{F}_0$. Recently, in ^[16], the authors introduced the generalized pseudo-packing measure ${\mathsf{R}}_{\mu}^{q, h}$ and they proved that

for all $A\subseteq \mathcal{X}$ and $B\subseteq \mathbb Y$, provided that we do not have $0$–$\infty$ case; that is, the product on the medium side does not take the form $0\times \infty$ or $\infty \times 0$. Note that we do not any restriction on the measures $\mu, \nu, h$, and $g$; that is, they do not satisfy necessary the doubling condition. In addition, one has (see again ^[16])

except in the $0$–$\infty$ case, where ${\mathsf{Q}}_{\mu}^{q, h}$ is the weighted generalized packing measure. In particular, one can obtain (1.1) under appropriate geometric conditions on $\mathcal{X}$ and $\mathcal{Y}$ (amenable to packing) ^[8,16].

Traditional packing and Hausdorff measures are defined using packings and coverings made up of collections of balls with diameters less than a given positive value $\delta$. An alternative approach to constructing fractal measures utilizes packings and coverings by using families of balls with a fixed diameter $\delta$. These measures, known as H-S measures, were first introduced in [17, Exercise (10.51)]. They were later explicitly described in Pesin's monograph ^[18] and are also referenced, albeit in an implicit manner, in foundational works such as Mattila's ^[19]. The importance of H-S measures goes beyond their theoretical definition; they offer a flexible framework for analyzing fractals and their complex characteristics. Numerous studies, including ^{[20,21,22,23]} for H-S measures and ^[24,25,26] for Standard measures, have demonstrated their utility in exploring the local properties of fractals and the behavior of fractal products. These works underscore the adaptability of H-S measures across various contexts, thus enriching the field of fractal geometry and its applications. Furthermore, Edgar's comprehensive exposition of these measures [27, pp. 32–36] provides a clear and accessible introduction, thoroughly detailing their construction, properties, and potential applications.

In Section 3, we are interested in studying the counterpart of the formula (1.1) related to the lower and upper H-S measures in Euclidean space. This result was shown for $q = 0$ in ^[28] in Euclidean space. We will prove the following theorem.

Theorem 1. Let $A\subseteq \mathbb R^d$, $B\subseteq \mathbb R^l$, $\mu\in \mathcal M_0(\mathbb R^d)$, $\nu\in \mathcal M_0(\mathbb R^l)$, $h, g \in \mathcal F_0$ and $q\in \mathbb R$. Positive constants $c_1$–$c_4$ exist such that

except in the $0$–$\infty$ case.

To prove the first inequality, we introduce a new multifractal measure that parallels the lower H-S measure and is notably simpler to analyze. This is achieved by utilizing a class of half-open dyadic cubes as covering sets in the definition, instead of using closed balls. The use of half-open dyadic cubes provides a new framework for the analysis, simplifying the structure of the measure. For the second inequality, we extend the technique by replacing the traditional dyadic cubes with half-open semi-dyadic cubes. This adjustment leads to the definition of two distinct measures that correspond to the upper and lower H-S measures. This choice arises from the fact that semi-dyadic cubes $v_n(x)$ are less sensitive to the position of $x$ compared with the corresponding dyadic cubes $u_n(x)$. Semi-dyadic cubes have been utilized in works such as ^[5,13,29]. It is important to note that this construction is specific to Euclidean space, making our proof distinct from those in ^[30].

Remark 2. It is important to emphasize that our analysis was not conducted for an arbitrary subset $\Gamma \subset {\mathbb R}^2$, but specifically for cases where $\Gamma$ takes the form of a Cartesian product

Addressing such a problem is far from straightforward, as it necessitates the application of integral versions of product set. For a deeper exploration of these techniques and their implications, we refer the reader to ^[5,31,32].

When

the measures $\mathsf{H}_{\mu}^{q, h}$ and $\mathsf{P}_{\mu}^{q, h}$ are simply denoted as $\mathsf{H}_{\mu}^{q, t}$ and $\mathsf{P}_{\mu}^{q, t}$, respectively. In this case, these measures assign, in the standard manner, a multifractal dimension to each subset $A$ of $\mathbb{R}^d$, defined as follows:

If $q = 0$, ${\mathsf b}_\mu(A)$ and ${\mathsf B}_\mu(A)$ do not depend on $\mu$ and are simply denoted ${\mathsf b}$ and ${\mathsf B}$, respectively. Theorem A implies, when all the hypothesis are satisfied, that

Moreover, all these inequalities may be strict. Indeed, one can construct two sets $A$ and $B$ such that

(see ^[33] for $q = 0$). However, in Example 2, we give a sufficient condition to get the first equality in Eq (1.5):

One can define also the multifractal separator functions

and

Where ${\mathsf b}_{\mu}$ is known to be a decreasing function, while ${\mathsf B}_{\mu}$ is both a decreasing and convex function ^[4]. In addition, it holds that

As a consequence, since

we get the following result:

by taking

and

in Theorem 1. Similar results were also proven for the $s$-dimensional Hausdorff measure $\mathcal H^s$ and the $s$-dimensional packing measure $\mathcal P^s$ ^[6,13,34,35]. In addition, a variety of related results and further developments on this problem can be found in the works of ^[36,37].

Now, given $\mu, \theta\in \mathcal{P} (\mathbb{R}^d)$, $q\in \mathbb R$, $h, g \in \mathcal F_0$, and $x\in\text{supp}(\mu)$, we define the upper and lower $(q, s)$-densities of $\theta$ at $x$ with respect to $\mu$ as

If

we use ${d}_{\mu}^{q, h}(x, \theta)$ to denote the common value. In ^[30], the authors used some density inequalities as "local versions" of the product inequalities. In particular, they proved that the inequality

may be deduced from the following density inequality:

where $\theta_1$ is the restriction of ${\mathsf P}^{q, h}_{\mu}$ to $E$ and $\theta_2$ is the restriction of ${\mathsf P}^{q, g}_{\nu}$ to $B.$

The set $A$ satisfies the condition

for measures $\mu$ under consideration, which will be called regular set. Regularity is defined with respect to various measures, such as the packing measure ^[29,38], the Hausdorff measure ^{[39,40,41,42]}, and the H-S measure ^[43,44,45]. Notably, Tricot et al. ^[38,46] demonstrated that a subset $A$ of $\mathbb{R}^d$ has integer Hausdorff and packing dimensions if it is strongly regular, meaning that

for $t \geq 0$. Furthermore, as a consequence of (1.5), it follows that if either $E$ or $F$ is regular, then

In Theorem 1, we assume that the products do not take the form $0 \times \infty$ or $\infty \times 0$. In Section 4, by estimating the measure of $d$-dimensional symmetric generalized Cantor sets, we demonstrate that this assumption is essential and can not be omitted. Specifically, let $0 < \alpha, \beta < 1$, to establish the second inequality in (1.4), we then need to prove that

for all

and some positive constant $c$, where $\overline {\mathsf H}^{q, \alpha}_{\mu, 0}$ and $\overline {\mathsf P}^{q, \beta}_\nu$ are the pre-lower and pre-upper H-S measures, respectively (see Section 3.2 and Eq (3.2)). We establish the following result.

Theorem 2. One-dimensional generalized Cantor sets $\mathcal K_1$, $\mathcal K_2^1$, $\mathcal K_2^2$, and $\mathcal K_2^3$ such that

and $\overline{ {\mathsf H}}_{\mu\times \nu, 0}^{q, \alpha+ \beta} (\mathcal K_1\times \mathcal K_2)$ and $\overline{ {\mathsf P}}_{\mu\times \nu}^{q, \alpha +\beta} (\mathcal K_1\times \mathcal K_2)$ are infinite, positive finite, and zero according as $j = 1, 2, 3$, respectively.

2.   Construction of fractal measures and preliminary results

2.1.   Construction of fractal measures

In this paper, we use formulas containing too many different variables, which is unpleasant, and omitting these extra parameters will create no confusion. To this end, for $\mu\in \mathcal P(\mathbb R^d), \nu\in \mathcal P(\mathbb R^l)$, and $h, g\in \mathcal F$, we define the gauge functions $\xi$ and $\zeta$ as

where $q\in \mathbb R$, $r > 0$, with the conventions

for $q\leq0$ and

for $q > 0$. The reader should note that we have simply used $\xi$ (respectively, $\zeta$) to denote the gauge function depending on $\mu$ (respectively, $\nu$), $q$, and $h$ (respectively, $g$). If

for $s, t \in \mathbb R$, then $\xi$ and $\zeta$ will be denoted as $\xi_s$ and $\zeta_t$ respectively. In this section, we construct the different fractal measures used in this paper. Let $\delta > 0$,

and $\{ \mathsf B(x_i, \; r_i)\}_i$ is a $\delta$-packing of the $A$, that is, a countable family of disjoint closed balls such that $x_i \in A$ and

for all $i$. Write

where the supremum is taken over all $\delta$-packings of the set $E$. The generalized packing measure $\mathcal P^{\xi}$ of $A$ with respect to $\xi$ is defined by

and

In a similar way, we define

where the infimum is taken over all $\delta$-coverings $\Big\{ \mathsf B(x_i, r_i)\Big\}_i$ of $E$; that is, $x_i\in E$, $0 < 2r_i < \delta$, and

We define the generalized Hausdorff measure as

and

We refer to ^[1,5] for more details (see also ^[46,47] for $q = 0$). Moreover, an integer $\kappa\in\mathbb{N}$ exists such that

Similarly, we define

where

It is clear that $\overline{\mathsf P}^{\xi}$ is increasing and

However it is not $\sigma$-additive. For this, we define the ${\mathsf{P}}^{\xi}$-measure defined as

In a similar way, we define

where

Clearly, $\overline{\mathsf H}_{0}^{\xi}$ is not countably subadditive and not increasing; one needs some modification to obtain an outer measure. More precisely, let

and

It is well known (see, for instance, ^[48]) that $\mathsf{H}^{\xi}$ and $\mathsf{P}^{\xi}$ are metric outer measures, which implies that they are measures on the Borel algebra. Moreover, for some integer $\kappa\in\mathbb{N}$, the following inequality holds:

2.2.   Construction of the generalized Cantor set

In the following, we recall the construction of the one-dimensional generalized Cantor set $\mathcal K$. Let ${\mathsf L}$ be a positive number, let $\{n_k\}_{k\ge 1}$ be a sequence of integers, and let $\{\lambda_k\}_{k\ge 1}$ be a sequence of positive numbers such that

for all $k\ge 1$. The construction of the generalized Cantor set $\Big\{ {\mathsf L}, \{n_k\}_{k\ge 1}, \{\lambda_k\}_{k\ge 1}\Big \}$ is as follows. In the first step, from a given closed interval with the length ${\mathsf L}$, remove $(n_1 - 1)$ open intervals and then leaves $n_1$ closed intervals with the length $\lambda_1$, denoted by $I_1, \ldots, I_{n_1}$. Let

In the second step, from each remaining closed interval with the length $\lambda_1$, remove $(n_2 - 1)$ open intervals and leaves $n_2$ closed intervals with the length $\lambda_2$. These are denoted as $I_{j_1, j_2}$, and we can write

We continue this process and, in the $k$-th step, obtain $n_1n_2 \cdots n_k$ closed intervals with the length $\lambda_k$, denoted $I_{j_1, j_2, \ldots, j_k}$ and denote their union as $J_k$. Then let

Let

be the uniform measure on $\mathcal K$, that is

and define

This construction can be generalized in $\mathbb R^d$ and $\mathcal K^d$, denoting the generalized Cantor set. Let $F_k$ be the product set of $d$ copies $J_k$. Thus, $F_k$ is the union of $(n_1n_2\ldots n_k)^d$ closed cubes with the side $\lambda_k$, each of which may be denoted as $Q^{(k)}$, and

The next lemma will be used in Section 4 to estimate the measure of $\mathcal K^d$.

Lemma 1. Let $\mathcal K^d$ be the $d$-dimensional symmetric generalized Cantor set ($d\geq1)$. A set function $\Psi$, defined on every non-empty closed subset in $\mathbb R^d$ and $r_0$, exists such that, for every open cube $I$ with the side $r\leq r_0$, we have

where $k$ is the unique integer such that

Proof. We start the proof by constructing the set function $\Psi$. Assume that

Let

then there is a $k_0$ such that

and

for all $k > k_0$. We define the sequence $({\lambda'}_k)$ such that

Clearly, we have $\lambda_k > \lambda'_k$ (since $h$ is increasing) and

Let $A$ be any open set and define

Then, we have

It follows that the sequence $\big\{ N_{\mu, k}^q (A) h(\lambda'_k) \big\}$ is decreasing, and we may define the function

Now, we will prove $(2.4)$. Let $I$ be an open cube, $k$ exists such that

Moreover, take a positive sequence $(\delta_k)_{k\ge 1}$ such that

Then the following exists:

such that

Observe that

It follows that

● If $j = 1$, then

● If $1 < j < n_{k+1}$, then

Since

and

is decreasing, we get

and then

Now, observe that

As a consequence, we obtain

This completes the proof. □

3.   Product formula: proof of Theorem 1

We set, for $n\in \mathbb N$,

and

The family $\mathcal U_n$ denotes the set of half-open dyadic cubes of order $n$. For $x \in \mathbb{R}^d$, let $\mathrm u_n(x)$ denote the unique cube $\mathrm u \in \mathcal U_n$ that contains $x$. Similarly, the family $\mathcal V_n$ consists of half-open dyadic semi-cubes of order $n$. For $x \in \mathbb{R}^d$, let $\mathrm v_n(x)$ represent the unique semi-cube $\mathrm v \in \mathcal V_n$ that contains $x$ and has its complement at a distance of $2^{-n-2}$ from $\mathrm u_{n+2}(x)$. Define

For each

let

Note that for

we have

Additionally, the collection $\big(\mathcal V_{ \textbf{k}, n}\big)_{ \textbf{k} \in \mathcal K}$ forms a partition of the family $\mathcal V_n$. Moreover, if

then either

or one is contained within the other. Finally, for $A \subset \mathbb{R}^d$, define

In what follows, we construct measures on $\mathbb{R}^d$ analogous to the generalized lower and upper H-S measures. However, instead of using the collection of all closed balls in the definition, we employ the class of all half-open dyadic semi-cubes. For $A \subseteq \mathbb{R}^d$, we define

where the numbers $N_n^*(A)$ and $M_n^*(A)$ are defined as

and

The functions $\overline{\mathsf{H}}^{*\xi}$ and $\overline{\mathsf{P}}^{*\xi}$ are increasing and satisfy

However these functions are not $\sigma$-additive. For this, we consider

Lemma 2. For every set $A\subset \mathbb R^d$, a constant $c > 0$ exists such that

Proof. This arises from the fact that

This completes the proof. □

Similarly, we may define ${\mathsf{H}}^{**\xi}$ and ${\mathsf{P}}^{**\xi}$, by using the class of all half-open dyadic cubes in the definition instead of the class of all half-open dyadic semi-cubes. However, it is important to note the resulting pre-measure, denoted $\overline{\mathsf{P}}^{**\xi}$, is not equivalent to the pre-measure $\overline{\mathsf{P}}^{\xi}$. For more discussion, consult [29, Example 3.5], where the interplay between the two pre-measures is explored. This highlights how seemingly minor changes in the class of sets used can lead to significant differences in the resulting pre-measures and their properties.

3.1.   Proof of the first inequality

In this section, for the sake of simplicity and clarity, we focus on results that pertain specifically to subsets of the plane. However, it is worth noting that these results can be extended to higher-dimensional spaces without significant complications. Let $\Pi \subset \mathbb{R}^2$ represent a subset of the plane. For a given $x$-coordinate, we use $\Pi_x$ to denote the set of all points in $\Pi$ whose abscissa ($x$-coordinate) equals $x$. Given an arbitrary subset $A$ of the $x$-axis, we will only prove that, if $x\in A$, we have

or some constant $a$, and then

Let $n$ be a non-negative integer and let $\{I_i\times I_j\}_{i, j}$ be a collection of half-open dyadic cubes of order $n$ covering $\Pi$. Set

Note that

This holds for any covering of $\Pi$ by the binary squares $\{I_i\times I_j\}_{i, j}$ with $2^{-n}$ sides. Hence,

Since $A_n$ increases to $A$ as $n \to +\infty$, then for any $p\le n$, we have

Thus, we obtain

for $p\ge 1$. Thereby, the continuity of the measure ${\mathsf H}^*$ implies that

Thus, using Lemma 2, we get

Finally, by taking

we get the result.

3.2.   Proof of the second inequality

Let $A\subseteq \mathbb R^d$ and $B\subseteq \mathbb R^l$. We prove that a constant $c > 0$ exists such that

Let

$r > 0$, and let $\{ \mathsf B(x_i, r)\}_i$ be a centered $r$-covering of $A$. We denote $n$ as the integer such that

For $\mathrm v\in \mathcal V_n(B)$ with

and each $i$, choose a point

and a point $y'_{i, \mathrm v}\in \mathrm v$ such that

Note that

As a consequence, we have the family $\big(\mathsf B((y_{i, \mathrm v}, y'_{i, \mathrm v}), 2r) \big)_{i\in \mathbb N, \mathrm v\in \mathcal V_n(B), \mathsf B(x_i, r)\times \mathrm v)\cap H\neq \emptyset}$, which forms a centered $(2r)$-covering of $H$. Furthermore, we get

for

It follows, for each $\textbf{k}\in \mathcal K$, that the family

is a centered $\eta_r$-packing of $B$. It follows that

Thus, by considering the infimum over all possible centered $r$-coverings of the set $A$, we get

Therefore,

where

Now, assume that

and

Then

It follows that

Since the cover $(A_i)$ of $A$ and the cover $(B_j)$ of $B$ were arbitrarily chosen, we obtain

This holds for all for all

which implies that

3.3.   Proof of the third inequality

Let

We aim to show that a constant $c > 0$ exists such that the following inequality holds:

For simplicity, we limit our discussion to subsets of the plane, although the result can be extended to higher dimensions without without significant complications. Let $\mathcal Q$ be any packing of $B$ consisting of semi-dyadic intervals, and let $\mathcal C$ be any covering of $A$ composed of semi-dyadic intervals. We define the following

Clearly, we have each of $\mathcal C_i$ is a packing of $E$ and $\mathcal C_i\times \mathcal Q$ is a packing of $A\times B$. Therefore,

This holds for any packing $\mathcal Q$ of $B$ and

so we have

Thus,

Finally, we get the desired result using (3.1).

3.4.   Proof of the fourth inequality

Let $A\subseteq \mathbb R^d$ and $B\subseteq \mathbb R^l$. We will prove that a constant $c > 0$ exists such that

Here again, we limit our study to subsets of the plane, since the extension to higher dimensions does not involve significant complications. Let $\mathcal B$ represent any packing of the set $A\times B$ containing semi-dyadic squares, where each square is formed as the Cartesian product of two semi-dyadic intervals. We define the sets as follows:

and

Next, we examine the subclasses

It is not difficult to note that each of $\mathcal C_1, \mathcal C_2$ is a packing of $A$ and, similarly, each of $\mathcal Q_1$, $\mathcal Q_2$ is a packing of $B$. Moreover, each square of the packing ${\mathcal B}$ is in the collection $\mathcal C_i\times \mathcal Q_j$, $i, j\in \{1, 2\}$. Therefore,

This holds, for any packing of $A\times B$, so we have

and then

Let

for

we have:

Since $(A_i)$ represents an arbitrary covering of $E$ and $(B_j)$ represents an arbitrary covering of $B$, we can deduce that

Finally, by applying (3.1), we obtain the desired conclusion.

3.5.   Applications of Theorem 1

Let $\mu, \theta\in \mathcal{M} (\mathbb{R}^d)$, $q, s, t \in \mathbb R$, and $x\in\text{supp}(\mu)$, and recall the upper and lower $(q, h)$-densities of $\theta$ at $x$ with respect to $\mu$ as defined in (1.7). In this section, we assume that

and

When studying fractal measures, a common question that naturally arises is whether we can guarantee the existence of subsets that possess finite or positive Hausdorff measures. This question becomes crucial in understanding the intricate structure of fractals, as it involves determining whether certain subsets exhibit measurable properties in terms of the Hausdorff measure, either finite or positive. Assume that

for some positive real number $\alpha$. The assumption (3.3) implies, for every $\delta > 0$ that is small enough, that

It follows that for

we then have

Letting $\delta$ tend to zero, we get

Note that the assumption (3.3) is satisfied; for instance, if we take

to be the Lebesgue measure with

In this case, we see that the Hausdorff measure constructed above is the standard Hausdorff measure $\mathcal H^\varphi$ with

Thus, for any closed nonempty set

every subset of $G$, including the empty set, is a subset of infinite measures. Thus, we may construct the measures ${ {\mathsf H}}^{\xi_s\zeta_t}$ for which the subset of finite measure properties can fail to hold for every closed set of infinite measures. One can assume also that for every $\delta > 0$, the following exists:

such that

Using Theorem 1, we formulate a sufficient condition to obtain

First, we will state the following result, which is a direct consequence of Theorem 1.

Corollary 1. Let $A\subset \mathbb R^d$ and $B\subset \mathbb R^l$, $\mu, \theta\in \mathcal {P}(\mathbb R^d)$, and $\nu, \theta'\in \mathcal {P}(\mathbb R^l)$ such that $\mu$ and $\nu$ satisfy the doubling condition. Let

such that

(1) Assume that if $\inf_{(x, y)\in G'} \underline{d}^{q, h_s}_\mu(x, \theta) < \infty$ and $\inf_{(x, y)\in G'} \underline{d}^{q, h_t}_\nu(x, \theta') > 0$, then ${\mathsf P}^{\xi_s\zeta_t} (G') < \infty$.

(2) Assume that if $\text{sup}_{(x, y)\in G'} \overline{d}^{q, h_s}_\mu(x, \theta) < \infty$ and $\text{sup}_{(x, y)\in G'} \overline{d}^{q, h_t}_\nu(x, \theta') > 0$, then ${\mathsf H}^{\xi_s\zeta_t} (G') > 0$.

Proof. Using [30, Lemma 3], we have

if

and

whenever

where $\gamma, \tilde \gamma$ are positive constants. for all $\theta\in \mathcal {P}(\mathbb R^d)$. Thus, the result follows from Theorem 1.

□

Example 1. Recall the construction of the Moran set given in Section 2.2.

Lemma 3. ^[49] Let $A \subset I$ be a Moran set that satisfies the strong separation condition, and let $\theta$ be a finite Borel measure with

Then there are some positive constants $c_i$ ($1 \leq i \leq 4$) depending on $\delta$ and $t$, such that the following inequalities hold for any $\varphi(i) \in A$:

Now consider the special case $I = [0, 1]$, $n_k = 2$, and $c_{kj} = \frac13$ for all $k\geq1$ and $1\leq j\leq n_k$. In this case, the Moran set $A = B$ is the classical ternary Cantor set. Let

and $\theta$ and $\theta'$ be probability measures on $I$ defined by

where

It is clear that

Moreover, we have

It follows, using Lemma 3, that

and

Corollary 1 implies that

Example 2. Let $\mu, \nu\in \mathcal M(\mathbb R)$, $q\in \mathbb R$, and let $A$ and $B$ be two sets of points in the $x$-axis and $y$-axis, respectively. In this example, we give a sufficient condition to obtain

From Theorem 1, we have

so we only have to prove the inverse inequality. For this, for $t, s \in \mathbb R$, we define the lower $\zeta_t$-dimensional density of $A$ at the point $y$ as

Fix $r > 0$ and define the set $I_y(r)$ as the centered interval on $y$ with the length $r$. For $n \ge 1$, consider the set

Assume that $D^{\zeta_t}(y) > 0$ for all $y\in F$, which implies clearly that $B_n \nearrow B$. In addition, if we prove that

for some $n\in \mathbb N$, then we deduce that

This gives the result if we choose

Now, we will prove (3.4). Let

Let $n$ be an integer and $0 < r \le 1/n$; we then define

We can extract f a finite subset $J(r)$ rom $I(r)$ such that $\widetilde B_n \subset J(r)$ and no three intervals of $J(r)$ have points in it.

Lemma 4. For $0 < r \le 1/n$, we have

Proof. Divide the set $J(r)$ into $J_1(r)$ and $J_2(r)$ such that in each of them the intervals do not overlap. Using the definition of the set $F_n$, we get

Similarly, we obtain

as required.

In the other hand, for $\epsilon > 0$, a sequence of sets $\{A_i\}$ exists such that

and that

Thus, we have a sequence $\{B_{i, j}\}$ of intervals of length $r$ covering $\widetilde A$ such that the family $\{B_{i, j}\}_j$, for each $i$, is a covering of $A_i$ and

Let $[a, b]$ represent any interval within the set $\{B_{i, j}\}$. Enclose all points in this set that fall between the lines $x = a$ and $x = b$ with squares whose sides are parallel to these lines. The projections of these squares onto the $y$-axis correspond to intervals in $J(r)$. In a similar manner, construct sets of squares for each interval in $\{B_{i, j}\}$, and denote the set of squares associated with the interval $[a, b]$ as $C(a, b)$. Since the number of squares in $C(a, b)$ does not exceed the number of intervals in $J(r)$, and each square intersecting $\widetilde A \times \widetilde B_n$ can be inscribed within a centered ball of diameter $r' = 3r$, it follows that:

Thus, using (3.5) and (3.6), we get

Since $\epsilon$ is arbitrarily, we get

Finally, we have

from which the Eq (3.4) follows. □

The result given in this example can be summarized in the next theorem.

Theorem 3. Let $E$ and $F$ be sets of points in $x$-axis and $y$-axis, respectively. Set

and assume that ${\mathsf H}^{\xi_s}(A), {\mathsf H}^{\zeta_t}(B)\in (0, \infty)$, and, for all $y\in F$, $D^{\zeta_t}(y) > 0$. In this case,

4.   Estimation of the measure of symmetric generalized Cantor sets

We define the set $\mathcal G$ of all continuous and increasing functions $h$ on $[0, t_0)$ for some $t_0 > 0$ satisfying $h(0) = 0$, and the function

is decreasing. We assume in this section that $h \in \mathcal G$ and that it t satisfies the doubling condition

4.1.   Estimation of the generalized Hausdorff measure

A cube $I(x, r)$ in $\mathbb R^d$ is a subset of the form

For a cube $I$, we use $l(I)$ to denote its side length. In this section, using cubes with sides of a length less than $\delta$ rather than closed balls, we define a generalized Hausdorff measure $\widetilde{ \mathcal H}_{\mu}^{q, h}$ equivalent to the generalized Hausdorff measure ${ \mathcal H}_{\mu}^{q, h}$. We prove that this measure is appropriate for estimating the measure of the generalized Cantor set. Let $\mu \in \mathcal P_{D}(\mathbb R^d)$, $h \in \mathcal{F}_0,$ and $q \in \mathbb R$. Define

where

with the infimum being taken over all coverings of $A$ by cubes with sides of a length $\leq \delta$. Then a constant $C$ exists such that

We will compute the estimation of the generalized Hausdorff measure of the $\mathcal K^d$. More precisely, we have the following result.

Theorem 4. Let $\mathcal K^d$ be the $d$-dimensional symmetric generalized Cantor set ($d\geq 1$) constructed by the system $\Big\{ {\mathsf L}, \{n_k\}_{k\ge 1}, \{\lambda_k\}_{k\ge 1}\Big\}$. We then have

Proof. We focus on proving only the left-hand inequality; the validity of the right-hand inequality can be established using similar argument. Let $\Psi$ be the set function in Lemma $1$. Let $\varepsilon$ be a positive number with $\varepsilon \leq r_0$ and $\{I_i\}$ be a $\varepsilon$-covering of $\mathcal K^d$ by open cubes with the sides $r_i\leq \varepsilon$. We have

Since $b$ is an arbitrary number such that

then we get the desired result. □

4.2.   Example: study of the case where $0$–$\infty$

In this example, we take $d = 1$ and we consider the one-dimensional generalized Cantor set $\mathcal K_1$ (resp. $\mathcal K_2$) constructed by the system $\Big\{ {\mathsf L}, \{n_k\}_{k\ge 1}, \{\lambda_k\}_{k\ge 1}\Big\}$ (resp. $\Big\{ {\mathsf L}, \{n_k\}_{k\ge 1}, \{\Lambda_k\}_{k\ge 1}\Big\}$) In the following, we consider $l = 1, d = 1, n_k = 2$, and

Theorem 5. The constants $M$ and $M'$ exist such that

Proof. We focus on proving only the left-hand inequality; the validity of the right-hand inequality can be deduced using the same idea. Assume that

otherwise, the result remains trivial. Let

and choose a positive integer $k_1$ satisfying the following inequality:

for all $k\ge k_1$. Now we define the sequence $(\tilde \Lambda_k)_{k\ge k_1}$ as

It follows that

for all $k\ge k_1$.

Let

and use $I_1^{k}$ (resp. $I_2^{k})$ to denote any of the closed intervals of the generation $r$ of $\mathcal K_1$ (resp. $\mathcal K_2$). Then

Note that

and then

It follows that the sequence $\big\{ N_{\mu\times \nu, k}^q (K) \lambda_k^\alpha \tilde\Lambda_r^\beta \big\}$ is decreasing, and we may define the function

Case $\beta\le \alpha$. We can choose $k_2 > k_1$ such that

Let

and consider any two-dimensional open cube $I$ with the side $s\le s_0$. Let $p$ and $k$ be the unique positive integers such that

Since

for $k \ge k_2$, we deduce that $k < p$. Moreover, the open cube $I$ meets at most $2^2$ rectangles of the form $I_1^{p}\times I_2^{k}$ and so meets at most $2^4$ rectangles of the form $I_1^{p+1}\times I_2^{k+1}$. Therefore, since $p > k$, it follows that

Since $2^k \Lambda_k^\beta$ decrease as $r$ increases, note that

Then,

□

Example 3. As a consequence, we construct an estimate of the generalized packing measures of product sets of one-dimensional generalized Cantor sets. Let $0 < \alpha$ and $\beta < 1$. In this example, we consider the one-dimensional generalized Cantor set $\mathcal K_1$ (resp. $\mathcal K_2$) constructed by the system $\Big\{ l, \{k_r\}_{r\ge 1}, \{\lambda_r\}_{r\ge 1}\Big\}$ $\left(\text{resp.}\; \Big\{ l, \{k_r\}_{r\ge 1}, \{\Pi_r\}_{r\ge 1}\Big\}\right)$. Set $l = 1$ and $n_k = 2$ and consider in the following:

(1) We have

Therefore, $\overline{ {\mathsf H}}_{\mu\times \nu, 0}^{q, hg} (\mathcal K_1\times \mathcal K_2)$ (resp. $\overline{ {\mathsf P}}_{\mu\times \nu}^{q, hg} (\mathcal K_1\times \mathcal K_2)$) is infinite, positive finite, and zero for $j = 1, 2, 3$, respectively.

(2) We have

Therefore,

(3) We have

Therefore,

5.   Conclusions and perspectives

Let $A \subseteq \mathbb{R}^d$ and $B \subseteq \mathbb{R}^l$. In this work, we present a novel approach that is distinct from that in ^[30], as it is specifically tailored for Euclidean spaces, to establish the following inequality:

This result holds under the assumption that $\mu, \nu, h, g$ satisfy the doubling condition and that none of the products is of the form $0 \times \infty$ or $\infty \times 0$. Furthermore, by analyzing the measures of symmetric generalized Cantor sets, we demonstrate that the exclusion of the $0$–$\infty$ condition is indispensable and thus cannot be omitted. Let $(\mathcal X, \rho)$ and $(\mathcal X', \rho')$ be two separable metric spaces. The result presented in this paper holds true for both $\mathcal X$ and $\mathcal X'$, though the approach used in our proof does not extend to metric spaces.

(1) Let ${\mathcal B}(\mathcal X)$ denote the family of closed balls in $\mathcal X$, and let $\Phi(\mathcal X)$ represent the class of pre-measures. A pre-measure is any increasing function

satisfying

It is natural to consider a general construction of $\mathcal H_\mu^{q, \xi}$, defined using a measure $\mu$ and a pre-measure $\xi$. Specifically, our result applies when

and allows for the choice

for all $\mathsf B \in {\mathcal B}(\mathcal X)$. Let

We define $\xi_0$, the Cartesian product measure generated from the functions $\xi$ and $\xi'$, on ${\mathcal B}(\mathcal X \times \mathcal X')$ as

We strongly believe that the resulting measure is particularly well-suited for studying Cartesian product sets. Under a suitable doubling condition, we obtain the following result:

for all $A \subset \mathcal X$ and $B \subset \mathcal X'$. This construction was first introduced by Kelly in ^[50]; see also ^[51].

(2) To establish the equality presented in Eq (5.1), we draw inspiration from the work of Kelly ^[50]. Specifically, we propose constructing a weighted lower H-W measure, denoted $\mathcal{W}_\mu^{q, h}$, for any given Hausdorff measure $h$. This approach involves assigning non-negative weights to the covering sets, adhering to what is commonly referred to as the third method for constructing an outer measure. On the basis of this framework, we conjecture that the equality in (5.1) holds if the constructed weighted measure satisfies

Similarly, one can construct a weighted upper H-W measure, denoted $\mathcal{Q}_\mu^{q, h}$, by following the same approach used for the weighted lower H-W measure but replacing covering with packing ^[8]. We conjecture that the equality

for all $A \subset \mathcal X$ and $B \subset \mathcal X'$, holds if the constructed weighted measure satisfies

(3) A similar result to (1.2) and (1.3) can be achieved by examining fractal pseudo-packings and weighted measures of the H-S type. The purpose of employing these generalizations is to eliminate the need for assuming the doubling condition.

(4) Our results in this paper can be readily extended to the setting of generalized lower and upper H-S measures, denoted ${\mathsf H}_{\mu_1\times \mu_2}^{q, h}$ and ${\mathsf P}_{\mu_1\times \mu_2}^{q, h}$. These fractal measures play a crucial role in the multifractal analysis of a measure relative to another measure ^[52].

Use of Generative-AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [KFU251033].

Conflict of interest

The author declares no conflict of interest.