# Fat Cantor Sets

I have been working on a problem for a while and ultimately I would like to prove that a certain function is absolutely continuous, strictly monotone increasing, and its derivative vanishes on a set of positive Lebesgue measure. Now, I thought I had this question solved but I was being naive. The function in question is constructed as follows, $$F \subseteq[0,1]$$ is a fat Cantor set with positive measure. Let $$E = [0,1] \setminus F$$, then for every $$x \in [0,1]$$ define $$f(x)= \int_0^x\mathbb{1}_E dt.$$ Now I think proving that this is absolutely continuous and that its derivative vanishes on a set of positive measure is independent of $$E$$ and $$F$$, please see here for an argument I gave earlier today to justify the latter. However proving that this is strictly monotone increasing is what is bothering me now. I had a mistake in my logic when working on this problem, but I cannot seem to form the correct argument. Perhaps here we do make use of the fat Cantor set.

Does anyone have any idea how to prove that $$f$$ is strictly monotone increasing?

Thank you so much, sorry for the many questions!

Krull.

The key here is that fat Cantor sets are nowhere dense; consequently, we have the following:

$$(*)\quad$$ Suppose $$0\le a. Then there is an entire interval contained in $$(a,b)\cap E$$; that is, there are $$c in $$(a,b)$$ such that $$(c,d)\subseteq E$$.

This implies that $$f$$ is strictly increasing: given $$x, by $$(*)$$ let $$c, d$$ be such that $$x and $$(c,d)\subseteq E$$. Then we can get a strictly positive lower bound on $$f(y)-f(x)$$ in terms of $$c$$ and $$d$$:

$$f(y)-f(x)=\int_x^y\mathbb{1}_Edx\ge \int_c^d\mathbb{1}_Edx=d-c>0.$$

So now all you have to do is prove $$(*)$$. This is an immediate corollary of the fact that $$F$$ is nowhere dense, which you should have proved already (it's one of the key facts about fat Cantor sets).

• @WolfgangKrull Since fat Cantor sets are closed, it's enough to show that the complement of a fat Cantor set is dense. Now think about the length of the "surviving intervals" in the construction of a fat Cantor set at stage $n$ as a function of $n$, and what this says about the density of its complement. May 3, 2021 at 1:07