Apparent Measure, Integral of Real Order, and Relative Dimension

1.1. Measure in ℝ

Let I_ℝ be the set of all bounded open intervals of ℝ, of the form I=]a,b[, where a≤b. In the case where a=b we have . We define the length by L(]a,b[)=b – a.

• Definition 1. For any x₀∈ℝ we call the length of order , the map , defined by

  
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• Remarks.

  • 1.

    If , the length

    

In one hand this length L is independent of the point x₀, it is intrinsic. On the other hand the length of real order for 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 x^α 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 , and the function can be considered as a density measure indeed:

  • a.

    If x₀≥b, we denote

    
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• b.

  If x₀≤a, we denote

  
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• c.

  If x₀∈[a,b], we have

  
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