Smith-Volterra-Cantor set
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Image:Smith-Volterra set.png In mathematics, the Smith-Volterra-Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real line R that is nowhere dense (in particular it contains no intervals), yet has positive measure.
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Construction
Similarly to the construction of the Cantor set, the Smith-Volterra-Cantor set is constructed by removing certain intervals from the unit interval [0, 1].
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
- <math>[0, 3/8] \cup [5/8, 1]</math>.
The following steps consist of removing subintervals of width <math>1/2^{2n}</math> from the middle of each of the <math>2^{n-1}</math> remaining intervals. So for the second step the intervals (5/32, 7/32) and (25/32, 27/32) are removed, leaving
- <math>[0, 5/32] \cup [7/32, 3/8] \cup [5/8, 25/32] \cup [27/32, 1]</math>.
Continuing indefinitely with this removal, the Smith-Volterra-Cantor set is then the set of points that are never removed.
Properties
By construction, the Smith-Volterra-Cantor set contains no intervals. During the process, intervals of total length
- <math> \sum_{n=0}^{\infty} 2^n(1/2^{2n + 2}) = 1/4 + 1/8 + 1/16 + \cdots = 1/2 \,</math>
are removed from [0, 1], showing that the set of the remaining points has a positive measure of 1/2.
Other fat Cantor sets
In general, you can remove rn 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 iff the sum of the sequence is finite.
See also
- The SVC is used in the construction of Volterra's function (see external link).
External links
- Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud