Cantor Function

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Cantor ternary Function

if is the Cantor set on [0,1], then the Cantor function c : [0,1] → [0,1] can be defined as[1]

Properties of Cantor Functions

  • Cantor Function is continuous everywhere, zero derivative almost everywhere.
  • lack of absolute continuity.
  • Monotonicity
  • Its value goes from 0 to 1 as its argument reaches from 0 to 1.

Cantor Function Alternative

The Cantor Function can be constructed iteratively using homework construction.[2]

Measurable function with pre-image of Lebesgue measurable set not Lebesgue measurable

Define


Then it can be shown is the pointwise limit of simple functions. However f takes values in the cantor set on the set of non terminating decimals. We can find a non measurable set such that is a null set and thus lebesgue measurable. Therefore fails to be Lebesgue measurable despite E being measurable.

This is analogous to the construction in HW5 2d) to 2 g) and is useful for motivating the definition of the Lebesgue measurable functions to be measurable


References

1. Terence Tao, An introduction to measure theory

  1. Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M. (2006). Expositiones Mathematicae. Elsevier BV. 24 (1): 1–37.
  2. Craig, Katy. MATH 201A HW 5. UC Santa Barbara, Fall 2020.