Given a Lebesgue measurable set, how to prove its image is Lebesgue measurable

Question

I know the image of a Lebesgue measurable set need not be Lebesgue measurable, but I can across this problem and it claimed that the following is true: If $X$ is a Lebesgue measurable and bounded subset of the reals, then $Y = \{x^3 + 2x : x \in X\}$ is also Lebesgue measurable. How do you prove this is true though?

Answer 1

The functions $f(x)=x^3+2x$ has bounded derivative since $X$ is bounded.

So $f$ is Lipschitz.

A Lipschitz function on the real line sends sets of measure zero to sets of measure zero and $F_{\sigma}$ sets to $F_{\sigma}$ sets

Thus it sends measurable sets to measurable sets since a measurable set is a union of an $F_{\sigma}$ set and a set of measure zero.

So $f(X)$ is measurable.

Answer 2

Another argument that does away with the assumption “$X$ is bounded” is the fact that $f:x \longmapsto x^3+2x$ is a homeomorphism of $\mathbb{R}$. So $f(X)=(f^{-1})^{-1}(X)$ is measurable.