About a set that is continuous image of a measurable set

Question

This is my first post. I hope that you can help me with a little hint. My problem says: If $S\subseteq \mathbb{R}$, $S^2$ is defined to be $S^2=\{s^2\ |\ s\in S \}$.Show that if $\lambda(S)=0$, then $\lambda(S^2)=0$, where $\lambda$ is Lebesgue measure.

I can prove measurability of $S^2$, but I can´t prove $\lambda(S^2)=0$. Can you give me any hint?

Thanks.

Answer 1

Hint: Actually if $f\in C^1(\mathbb R),$ then $m(S)=0\implies m(f(S))=0$ and the general result is no harder to prove. For the proof, WLOG $S\subset [-a,a]$ for some $a>0.$ Use the boundedness of $f'$ on $[-a,a]$ to show there is $C$ such that $m(f(I)) \le Cm(I)$ for each interval $I\subset [-a,a].$ Therefore ...