Apparent Measure, Integral of Real Order, and Relative Dimension

Apparent Measure, Integral of Real Order, and Relative Dimension

Faycal Adda (Community College of Qatar, Qatar)
DOI: 10.4018/978-1-7998-3122-8.ch013

Abstract

In this chapter, the author introduces a concept of apparent measure in R^n and associates the concept of relative dimension (of real order) that depends on the geometry of the object to measure and on the distance that separates it from an observer. At the end, the author discusses the relative dimension of a Cantor set. This measure enables us to provide a geometric interpretation of the Riemann-Liouville's integral of order αϵ├]0,1], and based on this interpretation, the author introduces a modification on the Riemann-Liouville's integral to make it symmetrical and then introduces a new fractional derivative that exploits at the same time the right and the left fractional derivatives.
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1. Measure Of Real Order

1.1. Measure in ℝ

Let I be the set of all bounded open intervals of ℝ, of the form I=]a,b[, where ab. In the case where a=b we have 978-1-7998-3122-8.ch013.m01. We define the length978-1-7998-3122-8.ch013.m02 by L(]a,b[)=b – a.

  • Definition 1. For any x0∈ℝ we call the length of order 978-1-7998-3122-8.ch013.m03, the map 978-1-7998-3122-8.ch013.m04, defined by

    978-1-7998-3122-8.ch013.m05
    (1)

  • Remarks.

    • 1.

      If 978-1-7998-3122-8.ch013.m06, the length

      978-1-7998-3122-8.ch013.m07

and if x0∈]a,b[, we have

978-1-7998-3122-8.ch013.m08

In one hand this length L is independent of the point x0, it is intrinsic. On the other hand the length of real order 978-1-7998-3122-8.ch013.m09 for 978-1-7998-3122-8.ch013.m10 depends on the distance that separates x0 and the boundaries of the interval ]a,b[. For example, for α∈]0,1[, the farther the point x0 is from the interval ]a,b[, the smaller the length of order α of this interval appears (thanks to the property of the function xα for α∈]0,1[). Because of this dependence of the position of the point x0, it can be called “apparent” length. It mimics the impression an observer gets for the measure of an object from his position located in the position x0: the object appears smaller to the observer when it gets further from him. The extension of this concept will be done later on.

  • 2.

    Obviously we have 978-1-7998-3122-8.ch013.m11, and the function can be considered as a density measure indeed:

    • a.

      If x0b, we denote

      978-1-7998-3122-8.ch013.m12
      (2)

with
978-1-7998-3122-8.ch013.m13
(3)
  • b.

    If x0a, we denote

    978-1-7998-3122-8.ch013.m14
    (4)

with
978-1-7998-3122-8.ch013.m15
(5)
  • c.

    If x0∈[a,b], we have

    978-1-7998-3122-8.ch013.m16
    (6)

where 978-1-7998-3122-8.ch013.m17 is a measure of the density 978-1-7998-3122-8.ch013.m18, it can be extended to any Borel set (“apparent” length of a Borel set).

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