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| [[Image:Smith-Volterra set.png|thumb|right|256px|After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2.]]
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| In [[mathematics]], the '''Smith–Volterra–Cantor set''' ('''SVC'''), '''fat Cantor set''', or '''ε-Cantor set'''<ref>Aliprantis and Burkinshaw (1981), Principles of Real Analysis</ref> is an example of a set of points on the [[real line]] '''R''' that is [[nowhere dense]] (in particular it contains no [[interval (mathematics)|interval]]s), yet has positive [[measure (mathematics)|measure]]. The Smith–Volterra–Cantor set is named after the [[mathematician]]s [[Henry John Stephen Smith|Henry Smith]], [[Vito Volterra]] and [[Georg Cantor]].
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| == Construction ==
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| Similar to the construction of the [[Cantor set]], the Smith–Volterra–Cantor set is constructed by removing certain intervals from the [[unit interval]] [0, 1].
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| The process begins by removing the middle 1/4 from the interval [0, 1] (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is
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| :<math>\left[0, \frac{3}{8}\right] \cup \left[\frac{5}{8}, 1\right].</math> | |
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| The following steps consist of removing subintervals of width 1/2<sup>2''n''</sup> from the middle of each of the 2<sup>''n''−1</sup> remaining intervals. So for the second step the intervals (5/32, 7/32) and (25/32, 27/32) are removed, leaving
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| :<math>\left[0, \frac{5}{32}\right] \cup \left[\frac{7}{32}, \frac{3}{8}\right] \cup \left[\frac{5}{8}, \frac{25}{32}\right] \cup \left[\frac{27}{32}, 1\right].</math>
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| Continuing indefinitely with this removal, the Smith–Volterra–Cantor set is then the set of points that are never removed. The image below shows the initial set and five iterations of this process.
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| [[Image:Smith-Volterra-Cantor set.svg|center|512px]]
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| Each subsequent iterate in the Smith–Volterra–Cantor set's construction removes proportionally less from the remaining intervals. This stands in contrast to the [[Cantor set]], where the proportion removed from each interval remains constant. Thus, the former has positive measure, while the latter zero measure.
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| == Properties ==
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| By construction, the Smith–Volterra–Cantor set contains no intervals and therefore has empty interior. It is also the intersection of a sequence of closed sets, which means that it is closed.
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| During the process, intervals of total length
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| :<math> \sum_{n=0}^{\infty} 2^n(1/2^{2n + 2}) = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = \frac{1}{2} \,</math>
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| are removed from [0, 1], showing that the set of the remaining points has a positive measure of 1/2. This makes the Smith–Volterra–Cantor set an example of a closed set whose [[Boundary (topology)|boundary]] has positive [[Lebesgue measure]].
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| == Other fat Cantor sets ==
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| In general, one can remove ''r''<sub>''n''</sub> from each remaining subinterval at the ''n''-th step of the algorithm, and end up with a Cantor-like set. The resulting set will have positive measure if and only if the sum of the sequence is less than the measure of the initial interval.
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| == See also ==
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| * The SVC is used in the construction of [[Volterra's function]] (see external link).
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| * The SVC is an example of a compact set that is not Jordan measurable, see [[Jordan measure#Extension to more complicated sets]].
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| * The indicator function of the SVC is an example of a bounded function that is not Riemann integrable on (0,1) and moreover, is not equal almost everywhere to a Riemann integrable function, see [[Riemann integral#Examples]].
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| == References ==
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| {{Reflist}}
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| ==External links==
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| * [http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf ''Wrestling with the Fundamental Theorem of Calculus: Volterra's function], talk by [[David Bressoud|David Marius Bressoud]]
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| {{DEFAULTSORT:Smith-Volterra-Cantor set}}
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| [[Category:Sets of real numbers]]
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| [[Category:Measure theory]]
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| [[Category:Topological spaces]]
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| [[Category:Fractals]]
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27 years old Plastic and Reconstructive Surgeon Crosser from West Hill, enjoys to spend time surfing, property developers new condo in singapore singapore and bee keeping. Likes to visit unfamiliar locations like Mapungubwe Cultural Landscape.