Lebesgue measure of a polynomial.

Question

Let $P:\mathbb{R} \rightarrow \mathbb{R}$ a polynomial. If $E \subset \mathbb{R}$ has measure zero, prove that $P(E)$ has measure zero.

Someone can help me? >-<

Answer 1

Let $n\in{\bf{N}}$ be fixed and $E_{n}=E\cap[n,n+1]$. We know that for some $M_{n}>0$, $|P'(x)|\leq M_{n}$ for all $x\in I_{n}$. Hence $\mu^{\ast}(P(E_{n}))\leq M_{n}\mu^{\ast}(E_{n})=0$, so $P(E_{n})$ is of measure zero.

Now $E=\displaystyle\bigcup_{n}E_{n}$ and hence $\mu^{\ast}(P(E))\leq\displaystyle\sum_{n}\mu^{\ast}(P(E_{n}))=0$.