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Currently I'm just working through measure theory just to see if its something I would like to take.

Unfortunately I am stuck on this problem from Carothers.

If $m^*(E)=0$, then $m^*(E^2)=0$.

Where $m^*$ denotes outer measure and $$E^2=\{x^2:x\in E\}.$$

I toyed with the idea that $I_k < 1 \Rightarrow I^2_k < I_k$. However I am at a loss as to how to set up a chain of inequalities (which is what I am assuming I need).

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    Hi W Rldt! Welcome to math.stackexchange. Before we can answer your question, could you please clarify what the square of a set is meant to mean here? I am uncertain and perhaps others will be as well.2011-09-15
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    This is the definition: $E^2=\{x^2 : x \in E\}$2011-09-15
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    What do you mean by $I_k<1$? Is $I_k$ an interval? If so, do you really mean that the length of $I_k$ is less than $1$? If so, note that this would not imply that $I_k^2$ has length less than $1$. E.g. $(5, 5.1)^2=(25,26.01)$.2011-09-15

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