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On a new measure on fractals
Journal of Inequalities and Applicationsvolume 2013, Article number: 522 (2013)
Abstract
Fractals are sets whose Hausdorff dimension strictly exceeds their topological dimension. The algorithmic Riemannianlike method, ${F}^{\alpha}$calculus, has been suggested very recently. HenstockKurzweil integral is the generalized Riemann integral method by using the gauge function. In this paper we generalize the ${F}^{\alpha}$calculus as a fractional local calculus that is more suitable to describe some physical process. We introduce the new measure using the gauge function on fractal sets that gives a finer dimension in comparison with the Hausdorff and box dimension. Hilbert ${F}^{\alpha}$spaces are defined. We suggest the selfadjoint ${F}^{\alpha}$differential operator so that it can be applied in the fractal quantum mechanics and on the fractal curves.
1 Introduction
Fractal geometry is used to describe real objects such as trees, lightning, river meanders and coastlines. We known that the Euclidean geometry is the approximate geometry applied in the real world. In the sense of Mandelbrot, a fractal set is the one whose Hausdorff dimension strictly exceeds the topological dimension [1–4]. Therefore, the calculus on fractals leads to better understanding the description of various real world models from science and engineering. Researchers have constructed analysis on fractals by using different approaches. For example, the fractional calculus has been applied to explain the supper and subdiffusion in physics as well as many other nonlocal phenomena. The fractional derivatives are nonlocal and they have memory property [5–8]. Recently, the fractional local calculus has been defined and applied in various fields [9–13]. The calculus on fractals which is Riemannianlike was suggested in [14, 15]. Meanwhile, it was generalized and applied in Newtonian, Lagrange and Hamilton mechanics. The Maxwell and the Schrödinger equations were expanded on fractal imbedding in ${R}^{3}$ [16–19]. Moreover, the measure theoretical approaches were used to create a calculus on fractals [20, 21]. In recent years, the calculus on fractals became an interesting and powerful tool for researchers. Taking into account the motivation presented above, in this manuscript we suggest a new measure on fractals. The relative extrema in ${F}^{\alpha}$calculus are discussed. The Hilbert ${F}^{\alpha}$spaces and ${F}^{\alpha}$selfadjoint differential operator are defined.
The plan of the paper is given below.
In Section 2 we review the definitions of measurable sets and Hausdorff measure. In Section 3 a new measure on fractal sets is introduced. We introduce the relative extrema condition of the ${F}^{\alpha}$calculus in Section 4 . Section 5 presents the Hilbert spaces in an ${F}^{\alpha}$space. In Section 6 the ${F}^{\alpha}$selfadjoint differential operator is defined. Section 7 is devoted to conclusions.
2 Preliminaries
As it is known, measure theory extends the concept of length for an arbitrary subset of the real line. Thus, a measure is the generalization of the concepts of length, area and volume [23]. Lebesgue introduced a measure that leads to a calculus on ${R}^{n}$. In this section we review the basic tools used in this paper. Firstly, we recall the definition of measurable sets such as Borel sets. Secondly, the Hausdorff measure is reviewed so that it is useful for analysis on fractals [23]. We notice that the Lebesgue measure is zero on the Cantor set as a fractal subset of the real line. More details about this subject can be found in the papers [9–15].
2.1 Measurable sets
Let R and $P(R)$ be all subsets or a power set of R. Boolean algebra ℜ of the sets in $P(R)$ has the following conditions [2, 3, 22, 23]:

(I)
$A,B\in \mathfrak{R}\Rightarrow A\cup B\in \mathfrak{R}$,

(II)
$A\in \mathfrak{R}\Rightarrow {A}^{c}=RA\in \mathfrak{R}$,

(III)
$A,B\in \mathfrak{R}\Rightarrow A\cap B\in \mathfrak{R}$,
where ℜ are subsets of a set $P(R)$. An algebra ℜ is σalgebra if it is closed under countable union of sets.
μ is a measure on ℜ if for a number of subsets of ℜ, we have
If ${A}_{1},{A}_{2},\dots $ is a countable sequence of sets, thus
equality is for the disjoint Borel sets.
2.2 Hausdorff measure
Suppose that the $(R,\rho )$ is a metric space and $K\subset R$, thus the diameter of K is [2, 3, 22, 23]
Consider Z to be a subset of R, then we conclude that
where $\delta >0$ and infimum is upon all countable covers of Z with the condition ${K}_{i}\subset R$ and $diam{K}_{i}<\delta $. The ${H}_{\delta}^{d}(S)$ is decreasing, so ${lim}_{\delta \to 0}{H}_{\delta}^{d}(S)$ exists. Then the following
is called Hausdorff measure on fractal sets [2, 3, 22, 23]. In this definition, the covering sets can be open or closed, and they will yield the same measure.
2.3 Generalized Hausdorff measure
We review the fractional measure introduced on fractional sets [9, 10].
The fractional sets are defined as follows:
The diameter of a nonempty fractional set E, the subset of a fractional set, is given by
If $\{{F}_{i}\}$ is a countable collection of subsets of ${R}^{\alpha}$ such as $E\subset {\bigcup}_{i=1}d{F}_{i}$ and $0<{F}_{i}<{\delta}^{\alpha}$, then $\{{F}_{i}\}$ is ${\delta}^{\alpha}$cover sets [9, 10]. The αdimensional generalized Hausdorff measure is suggested as follows:
In the following sections, we present the main results of our manuscript.
3 A new measure on fractals
In this section we write the definition of a flag function. By using the gauge function, the generalized mass function is defined so that it can be used for a wider class of functions on fractal sets. Let F be a subset of the real line and a fractal. The flag function for a set F has the form
where $I=[a,b]$ denotes an interval in R, and $P=\{{I}_{i},i=1,\dots ,n\}={\{{I}_{i}\}}_{i=1}^{n}$ is a tagged partition [22, 23]. Let $\delta (t)$ be a gauge function on I. So, we say that $\dot{P}$ is $\delta (t)$fine if
Let us assume that $I=[a,b]$ is a nonempty compact interval.
Let F be a subset of $I=[a,b]$. Let $\dot{P}$ be a δfine partition. ${\sigma}_{\ast}^{\alpha}[F,I]$ is defined as
where $a<b$ and $0<\alpha \le 1$.
Given the gauge $\delta (t)$ on $I=[a,b]$ and $a<b$, let $\dot{P}$ be a δfine partition. The coarse grained mass ${}^{\ast}\gamma _{\delta}^{\alpha}(F,a,b)$ of $F\cap [a,b]$ is given by
where $\dot{P}={max}_{1\le i\le n}({x}_{i}{x}_{i1})$. We take the infimum over all δfine partitions of I.
The mass function ${}^{\ast}\gamma ^{\alpha}(F,a,b)$ is defined as follows:
${}^{\ast}\gamma _{\delta}^{\alpha}$ is increasing with δ, then we can define [19, 22, 23]
if it exists. We mention that ${}^{\ast}\gamma _{0}^{\alpha}$ is not a measure because countable additivity fails. We define the new measure arising from the gauge function and denote it by ${\mathbf{G}}^{\alpha}(F,a,b)$. Let ${}^{\ast}\gamma ^{\alpha}(F,a,b)$ be a set of functions over Borel sets $F\subset R$ as
where ${F}_{i}$ are countable closed covers of the set F. Here, we are taking infimum over all closed covers of F. Now, as a measure ${\mathbf{G}}^{\alpha}(F,a,b)$, we check the properties as follows.
Properties Now, we check all the properties for the new measure, namely:
(1) It is clear that the measure of a null set is zero. Since ${}^{\ast}\gamma _{\delta}^{\alpha}(\mathrm{\varnothing},a,b)=0$, then ${\mathbf{G}}^{\alpha}(\mathrm{\varnothing},a,b)=0$.
(2) If $A\subseteq B$ and $B\subseteq {\bigcup}_{n\in N}{E}_{n}$, where ${E}_{n}$ is a measurable set, then the set A is also $A\subseteq {\bigcup}_{n\in N}{E}_{n}$. Subsequently, we have
Therefore we conclude that
(3) Now, we demonstrate that if $A={\bigcup}_{i\in N}{A}_{i}$, where ${A}_{i}$ is a sequence of measurable sets, then ${\mathbf{G}}^{\alpha}(A,a,b)\le {\sum}_{i=1}^{\mathrm{\infty}}{\mathbf{G}}^{\alpha}({A}_{i},a,b)$.
Proof If ${\sum}_{i=1}{\mathbf{G}}^{\alpha}({A}_{i},a,b)$ diverges, then there is nothing to prove. So, we assume that ${\sum}_{i=1}{\mathbf{G}}^{\alpha}({A}_{i},a,b)<\mathrm{\infty}$. Let $\u03f5>0$, then for each i there exist sets ${E}_{ni}$, $n\in N$, such that ${A}_{i}\subseteq {\bigcup}_{n=1}{E}_{ni}$ and
Since $A\subseteq {\bigcup}_{i}{\bigcup}_{n}{E}_{ni}$, we have
Since this holds for any $\u03f5>0$, therefore we obtain
(4) If $\{{A}_{i}\}$ is a countable family of disjoint measurable sets such that $A={\bigcup}_{i}{A}_{i}$, then
Proof Let ${A}_{1},{A}_{2},\dots ,{A}_{n}$ be disjoint measurable sets. Noticing that the set ${A}_{1}$ is measurable, if one picks out a test set as $T={A}_{1}\cup {A}_{2}\cup \cdots \cup {A}_{n}$, therefore, we have
Recalling that ${A}_{2}$ is measurable, by choosing a test set $T={A}_{2}\cup {A}_{3}\cup \cdots \cup {A}_{n}$, we have
In a finite number of steps, we conclude that
Notice that $\{{A}_{i}\}$ is a countable family of disjoint measurable sets. For each n, we have
Hence, we obtain the following inequality:
In view of (14)(18), the proof is complete. □
Finally, we define βdimension as it is given below
4 Relative extrema in ${F}^{\alpha}$calculus
In the rest of the paper, we generalize the ${F}^{\alpha}$calculus on a fractal subset of the realline [14]. We recall that the extrema of a function are the values that the function is either maximum or minimum. Suppose that $f:I\to R$ and $Sch(f)$ contained in F, which is αperfect [14], is said to have a relative maximum / relative minimum at $c\in I$ if there exists a neighborhood $V={V}_{\delta}(c)$ of c such that $f(x)\le f(c)$ [respectively $f(c)\le f(x)$] for all x in ${V}_{\delta}\cap F$. We say that f has a relative extremum at $c\in I$ if it has either a relative maximum or a relative minimum at c [19].
Analogue of Fermat’s theorem in Fα$F^{\alpha }$calculus Let $f:I\to R$ be a function such that $Sch(f)$ is contained in F which is an αperfect set. Suppose that $c\in I$ is a relative extrema of f. If f is ${F}^{\alpha}$differentiable at c, then ${D}_{F}^{\alpha}f(x)=0$ [14].
Proof Let c be a relative maximum. Then there is $\delta >0$ such that ${V}_{\delta}(c)=(c\delta ,c+\delta )\subset I$ for all $x\in {V}_{\delta}\cap F$, we have $f(x)\le f(c)$.
If ${D}_{F}^{\alpha}f(x)>0$ so that there exists a neighborhood ${V}_{\delta}(c)\subseteq I$ such that
If $x\in {V}_{\delta}\cap F$ and $x>c$, then we have
However, this contradicts that f has a relative maximum at c. Therefore, we cannot have ${D}_{F}^{\alpha}f(x)>0$. Similarly, we cannot have ${D}_{F}^{\alpha}f(x)<0$. Hence, we must have ${D}_{F}^{\alpha}f(x)=0$. □
Remark A similar proof applies if c is a relative minimum.
5 Hilbert spaces in ${F}^{\alpha}$calculus
The Hilbert space is a vector space equipped with inner product such that every Cauchy sequence has limit that belongs to the vector space. In most cases in the quantum mechanics, we need to construct the Hilbert spaces. Therefore, in this section, we introduce the analogues of Hilbert spaces on a fractal set subset of the real line [14, 19]. Consider the space of functions, $f:[a,b]\to R$ and $Sch(f)$ is an αperfect set[14]. Thus, we have the following:
This space is called a square ${F}^{\alpha}$integrable function and it is denoted by ${L}_{2}^{\alpha}$. If $f,g\in {L}_{2}^{\alpha}$, then $af+bg\in {L}_{2}^{\alpha}$, where $a,b\in R$. ${L}_{2}^{\alpha}$ is a complete vector space. We introduce the inner product in an ${F}^{\alpha}$space as
where $f,g\in {L}_{2}^{\alpha}$. This inner product gives us a positive definite norm for each vector such that
$\parallel f\parallel =0$ if only if $f=0$. A distance between two functions is given by
This inherits from the metric space ${L}_{2}^{\alpha}$ since this metric space is complete with respect to the metric defined in Eq. (28). Accordingly, ${L}_{2}^{\alpha}$ is an ${F}^{\alpha}$Hilbert space, which will also be denoted by ${\mathbb{H}}^{\alpha}$. A system of functions $\{{f}_{n}(x)\}$ in ${\mathbb{H}}^{\alpha}$ is said to be an orthogonal system if
Here, ${\delta}_{n,m}$ is the Kronecker delta. Let $\u3008{f}_{n}(x)\u3009$ be a sequence of functions which belongs to ${L}_{2}^{\alpha}$. Then there exists a function $f(x)\in {L}_{2}^{\alpha}$ with the following property:
Equivalently, one can say that ${f}_{n}(x)$ approaches $f(x)$ if for every $\u03f5>0$, there exists a number ${n}_{0}>0$ such that
The sequence of functions $\u3008{f}_{n}(x)\u3009$ belonging to ${L}_{2}^{\alpha}$ is called a Cauchy sequence if
In other words, for a given $\u03f5>0$, there exists a number ${n}_{0}>0$ such that
Equation (33) means that every Cauchy sequence on a fractal set has limit belonging to the vector space. Finally, we have complied with all the conditions for a Hilbert space in ${F}^{\alpha}$calculus.
6 Selfadjoint ${F}^{\alpha}$differential operator
It is interesting to note that a secondorder differential operator can be written in a SturmLiouville form if it is selfadjoint. In quantum mechanics, the eigenvalues and eigenvectors are the energy and the wave function, respectively. So, by extending the ${F}^{\alpha}$calculus, namely defining an ${F}^{\alpha}$differential operator, we gain the definition and the condition for the analogous Hilbert operator on a fractal set. In this section we present the selfadjoint operators involving the fractional local derivative on fractals.
Suppose that L is a fractional local differential operator in the form
Now, we would like to calculate
By using integration by parts [14] once or twice, we conclude that
Comprising the integrals in Eqs. (36) and (35), we get
and
Then, we finally conclude that
When $\overline{\mathbf{L}}[f(x)]=\mathbf{L}[f(x)]$, the operator is said to be ${F}^{\alpha}$selfadjoint operator [19]. It is clear that every ${F}^{\alpha}$selfadjoint operator can be written as an analogous SturmLiouville equation on fractal sets.
7 Conclusion
Researchers are trying to create new calculus on fractals by using some different approaches which are algorithmic or not. Since the fractals generated are different, the measures on them also vary from one to another. The gauge function is used to generalize the Riemann integral for a wider class of functions. We defined a new measure on fractals by using this function. This new measure can be used to define an integral on fractals. Meanwhile, the ${F}^{\alpha}$calculus on fractals was generalized by given the relative extrema condition. Moreover, the Hilbert ${F}^{\alpha}$space is constructed for potential applications on fractal quantum mechanics and the selfadjoint ${F}^{\alpha}$differential operator is defined on fractal sets.
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Acknowledgements
One of the authors (AKG) would like to thank Professor AD Gangal for useful discussion on this topic during the period of time he was in Pune University.
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Keywords
 fractal measure
 fractal calculus
 fractal curve