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#### TheBigBadBen

##### Active member

- May 12, 2013

- 84

At any rate, two related questions:

(1)

Suppose that \(\displaystyle E \subset \mathbb{R}\) is a set such that \(\displaystyle m^*(E)=0\). Prove that \(\displaystyle m^*(E^2)=0\), where \(\displaystyle E^2 = \{x^2|x\in E\}\)

(2)

Suppose that \(\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}\) is a K-Lipschitz function. Show that \(\displaystyle m^*(E^2)≤Km^*(E)\) for all \(\displaystyle E\subset\mathbb{R}\)

Note that \(\displaystyle m^*\) refers to the Lebesgue outer-measure.