0
$\begingroup$

Let $ X \subset [0,1]$ and $f:X \rightarrow C$ be an injective mapping into a Cantor Set $C$. How do I justify whether $f(X)$ is Lebesgue measurable or not?

The Cantor set $C$ has measure $0$ and since the mapping is injective $X = f^{-1}(C)$ has measure $0$.

  • 0
    Isn't the Lebesgue measure of the Cantor Set $0$?2012-11-21

2 Answers 2

3

Every subset of a measure zero set is Lebesgue measurable.

2

What you mean to say is that $f(X)$ has measure $0$ because the cantor set has measure $0$ and lebesgue measure is complete.

  • 0
    @BhavishSuarez Glad I could help!2012-11-21