If E has positive measure, does $E^2$ have positive measure?

Question

I know the statement " If $E$ is a subset of real numbers with measure zero,then so is $E^2$={$x^2$: $x\in E$} " is true. How about " If $E$ is a subset of real numbers with positive measure, then so is $E^2$={$x^2$: $x\in E$} " ? Is true or false ?

Answer 1

If $E$ has positive measure, then at least one of $E\cap\mathbb{R}_+$ and $E\cap\mathbb{R}_-^\ast$ has positive measure. Assume without loss of generality that $E\cap\mathbb{R}_+$ has positive measure. Then $(E\cap\mathbb{R}_+)^2$ must have positive measure as $f\colon x\in\mathbb{R}_+\mapsto x^2$ is a differentiable bijection.

But $(E\cap\mathbb{R}_+)^2\subseteq E^2$, so $E^2$ also has positive measure.