Uniform distribution on the Cantor set ( Durrett's probability : theory and examples)

Question

I"m trying to understand why the uniform distribution on the Cantor set cannot have a density function. Durrett's probability : theory and examples (https://services.math.duke.edu/~rtd/PTE/PTE4_1.pdf) has an example (Example 1.2.4) which proves that.

1. In that the author says that $f$ (the required density function) would be zero on a set of measure 1 hence there is no such $f$. Is it correct when I understand that the author is referring to the Cantor set whose measure is 1? Why can there exist no such $f$ though?
2. The author also says that it is an example of a singular distribution. Following definition 6.19 here (https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch6.pdf) since $F$ is piecewise constant on the disjoint portions of $C^C \implies \mu(C^C) = 0$ and since the lebesgue measure of $C$ is zero (since the cantor set comprises uncountably many points and the lebesgue measure of a point is zero), $\mu$ is a singular measure. Is this correct?

Answer 1

Several mistakes in your post:

• 1) note that the Cantor set has measure $0$ (and not $1$). Thus the complement (in $[0,1]$) of the Cantor set has measure $1$, from which you get that the density function is $0$ almost everywhere. Thus $\displaystyle{\int_{\mathbb{R}}} f(x)dx=0$, but it should be $1$: absurd

  • 2) you said that the lebesgue measure of $C$ is zero because it has uncountably many points and a point has Lebesgue measure zero. This is plain wrong: a countable union of points (i.e. a countable set has measure zero). $[0,1]$ is an uncountable union of points but has not measure zero, you cannot sum uncountably many measures. What you should write to explain why $F$ is singular is that outside the Cantor set, $F$ is locally constant on $C^C$, which is an open set. Thus $F$ is differentiable with derivatve zero on $C^C$, i.e. outside a negligeable set. Moreover, $F$ is non-decreasing and non constant. This is the definition of a singular function.