The Cantor set and integrability of $\frac{1}{x}$

Question

Let $\chi_C$ be the characteristic function of the standard Cantor set fully contained in the interval $[0,1]$. The problem is to resolve if $\lim\limits_{\varepsilon\to 0^{+}} \int_{\varepsilon}^1\frac{\chi_C}{x}dx$:

a) doesn't exist

b) is equal to $+\infty$.

c) exists.

Note: If there is computed $\int_{\varepsilon}^1\frac{\chi_C}{x}dx$ for $\varepsilon>0$, then please use R-integration.

Answer 1

Note that $f$ is bounded on $[\epsilon,1]$, so you just need to show that the integral of a bounded function which vanishes outside the Cantor set is zero. Use the fact that the Cantor set is covered by a finite union of closed intervals of arbitrarily small measure.

Answer 2

The standard Cantor set has Lebesgue measure zero, so you just have to integrate $0$.