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If $E$ has measure zero, then does $E^2$ have measure zero?
I'm trying to find a proof for the following question: Suppose $A\subset\mathbb{R}$ is a set of measure zero. Show that the set $A^2=\{x^2 \in \mathbb{R} \,|\, x \in A\}$ is also a set of measure zero.
I feel like this is a very easy proof and I am just missing something. I don't think stating "the map $f:A\to A^2$ also maps the collection of intervals covering $A$ to a collection of intervals covering $A^2$" is complete. I don't believe this qualifies as a Lipschitz function, since the derivative of the function $f$ is unbounded on $\mathbb{R}$. Please help, is there something I am missing here?