Smith-Volterra-Cantor crowd

After removing all black marked intervals, the remaining points form a nowhere dense set with Lebesgue measure 1/2.

Under the Smith-Volterra-Cantor set (short SVCM ) or fat Cantor set is understood in mathematics, a subset of the unit interval that nowhere dense (thus contains in particular no interval), but still a strictly positive Lebesgue measure has . The crowd was named after the three mathematicians Henry Smith , Vito Volterra and Georg Cantor . The SVCM is homeomorphic to the (“lean”) Cantor set .

definition

Like the Cantor set, the SVCM is constructed by removing intervals from the unit interval . ${\displaystyle [0,1]}$

You start by removing the middle quarter from the unit interval. The remaining amount after the first step is thus

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

At the -th step, an interval of length is now removed from the center of each remaining interval. After the second step, for example, the following amount remains: ${\displaystyle n}$ ${\displaystyle {\frac {1}{4^{n}}}}$

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

The average of all the amounts after each step is the amount of points that are never removed. That crowd is the SVCM. The first five steps are visualized in the picture below:

Each step of the process removes a proportionally smaller portion of each interval than the previous step. This is different from the method of constructing the Cantor set, in which each step removes one third of each interval.

The formal definition is as follows: Define . For every natural number n let (inductive) ${\displaystyle S_{0}=[0,1]}$

${\displaystyle S_{n}:=\bigcup _{k=1}^{2^{n-1}}\left(\left[a_{k},{\frac {a_{k}+b_{k}}{2}}-{\frac {1}{2^{2n+1}}}\right]\cup \left[{\frac {a_{k}+b_{k}}{2}}+{\frac {1}{2^{2n+1}}},b_{k}\right]\right)}$,

being the through ${\displaystyle 0=a_{1}<b_{1}<a_{2}<b_{2}<\dots <a_{2^{n-1}}<b_{2^{n-1}}=1}$

${\displaystyle S_{n-1}=\bigcup _{k=1}^{2^{n-1}}[a_{k},b_{k}]}$

are clearly given.

The SVCM is now . ${\displaystyle \bigcap _{n\in \mathbb {N} }S_{n}}$

properties

By definition, the SVCM does not contain an interval and therefore has an empty interior . As the average of closed sets, the SVCM is also closed. Since in all steps together a quantity is removed from the unit interval which is the Lebesgue measure

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

the SVCM has a Lebesgue measure of . In particular, the SVCM is an example of a set whose edge has a strictly positive Lebesgue measure. ${\displaystyle {\frac {1}{2}}}$

Other fat Cantor sets

The construction does not necessarily have to remove intervals of length every time . In general, in the -th step, intervals of length can be removed from any remaining interval, being any sequence of positive real numbers. The resulting Cantor-like set has a strictly positive measure if and only if the sum of the lengths of all distant intervals is less than the length of the start interval. ${\displaystyle {\tfrac {1}{4^{n}}}}$ ${\displaystyle n}$ ${\displaystyle c_{n}}$ ${\displaystyle c_{n}}$ ${\displaystyle \textstyle \sum _{n=1}^{\infty }2^{n-1}\cdot c_{n}}$

If, for example, a choice is made, then the resulting set has a strictly positive measure if and only if applies. ${\displaystyle c_{n}=a^{n}}$ ${\displaystyle a<{\tfrac {1}{3}}}$

Cartesian products from SVMCs can be used to find totally disjointed spaces with strictly positive dimensions in higher dimensions.

See also

• The SVCM is used in the construction of the Volterra function .
• The SVCM is a compact set that is not Jordan measurable .
• The indicator function of the SVCM is a restricted, non- Riemann-integrable function on the unit interval. In addition, there is no Riemann-integrable function that is almost everywhere the same as the SVCM's indicator function.

Individual evidence

1. ^ Bressoud, David Marius (2003). Wrestling with the Fundamental Theorem of Calculus: Volterra's function , speech by David Marius Bressoud .
2. ↑ The Smith Volterra Cantor Set | Math counterexamples. Retrieved October 7, 2019 (American English).