Exercise 1.2.27 (Projections of Lebesgue measurable sets need not be Lebesgue measurable)

Question

Let $\pi:\mathbb{R}^2	o\mathbb{R}$, $(x,y)\mapsto x$ be the two dimensional projection function. Show that there exists a Lebesgue measurable set $E \subseteq \mathbb{R}^2$ such that $\pi(E)$ is not Lebesgue measurable.

Elu Thingol · Accepted Answer

Consider the non Lebesgue measurable \href{./proposition_1-2-18}{Vitali set} $V$ from the Proposition 1.2.18. Then the set $E:=V 	imes \{0\}$ is a Lebesgue null set, and thus must be Lebesgue measurable by Lemma 1.2.13. It's projection $\pi(E)=V$, however, is the \href{./proposition_1-2-18}{Vitali set} itself, and is thus not measurable.

Exercise 1.2.27 (Projections of Lebesgue measurable sets need not be Lebesgue measurable)

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Exercise 1.2.27 (Projections of Lebesgue measurable sets need not be Lebesgue measurable)

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