Smith–Volterra–Cantor set

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) 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. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor.

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].

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

$\left[0,{\frac {3}{8}}\right]\cup \left[{\frac {5}{8}},1\right].$

The following steps consist of removing subintervals of width 1/2²ⁿ from the middle of each of the 2ⁿ⁻¹ remaining intervals. So for the second step the intervals (5/32, 7/32) and (25/32, 27/32) are removed, leaving

$\left[0,{\frac {5}{32}}\right]\cup \left[{\frac {7}{32}},{\frac {3}{8}}\right]\cup \left[{\frac {5}{8}},{\frac {25}{32}}\right]\cup \left[{\frac {27}{32}},1\right].$

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.

Properties

By construction, the Smith–Volterra–Cantor set contains no intervals. During the process, intervals of total length

$\sum _{n=0}^{\infty }2^{n}(1/2^{2n+2})={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots ={\frac {1}{2}}\,$

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, one can remove r_n 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.

See also

• The SVC is used in the construction of Volterra's function (see external link).
• The SVC is an example of a compact set that is not Jordan measurable, see Jordan measure#Extension to more complicated sets.
• 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.

External links

• Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud