Math 2.0

Measure and integration Construction of measures Example

Smith–Volterra–Cantor set

Builds on Lebesgue measureClosed setCantor set

Example

Start with C0=[0,1]C_0 = [0, 1]. At step n≥1n \ge 1, the set Cn−1C_{n-1} is a union of 2n−12^{n-1} disjoint closed intervals. Remove from the middle of each of them an open interval of length 4−n4^{-n}, and call the result CnC_n. The Smith–Volterra–Cantor set is S=⋂n≥0CnS = \bigcap_{n \ge 0} C_n. Then SS is a closed set, it is nowhere dense in ℝ\R, and its Lebesgue measure is 1/21/2.

Remarks

Step 11 removes (3/8,5/8)(3/8, 5/8) and leaves [0,3/8]∪[5/8,1][0, 3/8] \cup [5/8, 1]. By induction, each of the 2n2^n intervals of CnC_n has length 2−n−1+2−2n−12^{-n-1} + 2^{-2n-1}. This is more than 4−n−14^{-n-1}, so each removal is possible. The total length removed is ∑n=1∞2n−1⋅4−n=14∑n=0∞2−n=12,\sum_{n=1}^{\infty} 2^{n-1} \cdot 4^{-n} = \frac{1}{4} \sum_{n=0}^{\infty} 2^{-n} = \frac{1}{2}, so λ(S)=1−1/2=1/2\lambda(S) = 1 - 1/2 = 1/2. The set SS is closed as an intersection of closed sets. These lengths tend to 00, so SS contains no interval; a closed set with empty interior is nowhere dense. So a nowhere dense set can have positive measure, unlike the middle-thirds Cantor set, which has measure 00. The indicator function of SS is bounded and discontinuous at every point of SS, a set of positive measure, so it is not Riemann integrable.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptCantor set, example, no evidence levelCantor setexampleClosed set, concept, no evidence levelClosed setconceptEuclidean space, object, no evidence levelEuclidean spaceobjectOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectSmith–Volterra–Cantor set, example, no evidence levelSmith–Volterra–Cantor…example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Cantor set example
  3. Closed set concept
  4. Euclidean space object
  5. Outer measure concept
  6. Lebesgue measure object
  7. Smith–Volterra–Cantor set example

Further reading

  1. Web https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set
About this page
TagHZTRH10Z8Y
URIm2://HZTRH10Z8Y
Kindexample
WritingW0 stub
MSC28A05, 26A30
Instance ofNowhere dense set
UsesLebesgue measure, Closed set, Cantor set
Same as
Changed2 November 2026

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