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Smith–Volterra–Cantor set

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(Redirected from Smith-Volterra-Cantor Set) Set of real numbers in mathematics
After black intervals have been removed, the white points which remain are a nowhere dense set of measure 1/2.

In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881. The Cantor set as we know it today followed in 1883. The Smith–Volterra–Cantor set is topologically equivalent to the middle-thirds Cantor set.

Construction

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 ] . {\displaystyle .}

The process begins by removing the middle 1/4 from the interval [ 0 , 1 ] {\displaystyle } (the same as removing 1/8 on either side of the middle point at 1/2) so the remaining set is [ 0 , 3 8 ] ∪ [ 5 8 , 1 ] . {\displaystyle \left\cup \left.}

The following steps consist of removing subintervals of width 1 / 4 n {\displaystyle 1/4^{n}} from the middle of each of the 2 n − 1 {\displaystyle 2^{n-1}} remaining intervals. So for the second step the intervals ( 5 / 32 , 7 / 32 ) {\displaystyle (5/32,7/32)} and ( 25 / 32 , 27 / 32 ) {\displaystyle (25/32,27/32)} are removed, leaving [ 0 , 5 32 ] ∪ [ 7 32 , 3 8 ] ∪ [ 5 8 , 25 32 ] ∪ [ 27 32 , 1 ] . {\displaystyle \left\cup \left\cup \left\cup \left.}

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.

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 Smith–Volterra–Cantor set has positive measure while the Cantor set has zero measure.

Properties

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. During the process, intervals of total length ∑ n = 0 ∞ 2 n 2 2 n + 2 = 1 4 + 1 8 + 1 16 + ⋯ = 1 2 {\displaystyle \sum _{n=0}^{\infty }{\frac {2^{n}}{2^{2n+2}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots ={\frac {1}{2}}\,} are removed from [ 0 , 1 ] , {\displaystyle ,} 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 has positive Lebesgue measure.

Other fat Cantor sets

In general, one can remove r n {\displaystyle r_{n}} from each remaining subinterval at the n {\displaystyle 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. For instance, suppose the middle intervals of length a n {\displaystyle a^{n}} are removed from [ 0 , 1 ] {\displaystyle } for each n {\displaystyle n} th iteration, for some 0 ≤ a ≤ 1 3 . {\displaystyle 0\leq a\leq {\dfrac {1}{3}}.} Then, the resulting set has Lebesgue measure 1 − ∑ n = 0 ∞ 2 n a n + 1 = 1 − a ∑ n = 0 ∞ ( 2 a ) n = 1 − a 1 1 − 2 a = 1 − 3 a 1 − 2 a {\displaystyle {\begin{aligned}1-\sum _{n=0}^{\infty }2^{n}a^{n+1}&=1-a\sum _{n=0}^{\infty }(2a)^{n}\\&=1-a{\frac {1}{1-2a}}\\&={\frac {1-3a}{1-2a}}\end{aligned}}} which goes from 0 {\displaystyle 0} to 1 {\displaystyle 1} as a {\displaystyle a} goes from 1 / 3 {\displaystyle 1/3} to 0. {\displaystyle 0.} ( a > 1 / 3 {\displaystyle a>1/3} is impossible in this construction.)

Cartesian products of Smith–Volterra–Cantor sets can be used to find totally disconnected sets in higher dimensions with nonzero measure. By applying the Denjoy–Riesz theorem to a two-dimensional set of this type, it is possible to find an Osgood curve, a Jordan curve such that the points on the curve have positive area.

See also

References

  1. Aliprantis and Burkinshaw (1981), Principles of Real Analysis
  2. Smith, Henry J.S. (1874). "On the integration of discontinuous functions". Proceedings of the London Mathematical Society. First series. 6: 140–153
  3. Ponce Campuzano, Juan; Maldonado, Miguel (2015). "Vito Volterra's construction of a nonconstant function with a bounded, non Riemann integrable derivative". BSHM Bulletin Journal of the British Society for the History of Mathematics. 30 (2): 143–152. doi:10.1080/17498430.2015.1010771. S2CID 34546093.
  4. Balcerzak, M.; Kharazishvili, A. (1999), "On uncountable unions and intersections of measurable sets", Georgian Mathematical Journal, 6 (3): 201–212, doi:10.1023/A:1022102312024, MR 1679442.
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