# Measure of the reciprocal of a Cantor set

I have recently started studying measure theory and as is usual we started out by calculating the measure of the Cantor set. Now I had this question in my mind as to whether the set generated by taking the reciprocals of the elements of the Cantor set would be measurable or not!( I am removing the zero from the Cantor set here. Would that create some sort of a problem?)

So I started thinking about the operation of reciprocation. Now if I consider my set to be the open interval $(0,1)$, then its Lebesgue measure would be 1 and the set generated by taking the reciprocals of the elements in this set, i.e. the open set $(1, \infty)$ would have the measure $\infty$. This led me to conclude that such an operation might not behave well with the idea of Lebesgue measure.

This led to a further query regarding the measure of a set which has a bijective mapping to a measureable set. In other words, say $f: X \rightarrow Y$ and that the measure of $X$ is known, then can we say anything about the measure of $Y$.

I have only just begun the study of measure theory so please excuse the impreciseness of my statements.

NOTE: By measure I mean the Lebesgue measure in this question and by Cantor set I mean the one obtained by repeatedly removing the middle one-thirds of the intervals from the set $[0,1]$.

## 1 Answer

Let $C$ be the Cantor ternary set. For any $0 < a < 1$, consider $X_a = \{1/x : x \in C, x > a\}$. It is enough to show that each $X_a$ has measure zero but this is obvious because $f(x) = 1/x$ is Lipschitz on $[a, 1]$ (for every $a < x <y < 1$, $|f(x) - f(y)| < \frac{1}{a^2}|x - y|$).

Two more exercises for you: What are the measures of $C + C = \{x + y : x, y \in C\}$ and $C .C = \{xy: x, y \in C\}$.