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Let $E\subseteq\Bbb R$ be measurable with positive Lebesgue measure. Let $E_1,E_2$ be the projections of $E$ on the $x$ - axis and $y$ - axis respectively. Can we say that $E_1,E_2\subseteq\Bbb R$ are measurable with positive Lebesgue measures?

Matcha Latte
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Prof.Hijibiji
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1 Answers1

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Projections need not be measurable, but if they are measurable then their measures are necessarily positive: if $A$ and $B$ are the projections of $E$, then $E \subset A\times B$ and the measure of $A\times B$ would be $0$, if one of the projections has measure $0$.

Counterexample: let $A$ be any non-measurable set in $\mathbb R$ and $E=A\times \{0\}$. Then $E$ is Lebesgue measurable but its first projection is not. If you want to get an example where $E$ has positive measure but one of its projections is not measurable, just take the union with a disjoint rectangle; for example, we can take $A$ to be a non-measurable subset of $[0,1]$ and consider $(A\times \{0\}) \cup ([2,3]\times [2,3])$.

VacciChien
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Kavi Rama Murthy
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    Sir please note that in my question $E$ is of positive measure in $\mathbb{R}^2$. But in your example measure of $E$ is zero. – Prof.Hijibiji Mar 21 '19 at 07:10
  • If $E\subseteq R^2$ is Borel measurable, is it still true that its projections may be non-measurable? – Jack M Mar 21 '19 at 07:14
  • I have edited the answer to get a set of positive measure. Projection of a Borel set need not be measurable. But no simple concrete example can be given. @Prof.Hijibiji – Kavi Rama Murthy Mar 21 '19 at 07:20
  • @Prof.Hijibiji See https://math.stackexchange.com/questions/603173/a-borel-set-whose-projection-onto-the-first-coordinate-is-not-a-borel-set – Kavi Rama Murthy Mar 21 '19 at 07:23
  • Thank you @Kavi Rama Murthy sir. After your edited answer I am provoked to ask, whether $E_1,E_2$ contains a positive measure subset? – Prof.Hijibiji Mar 21 '19 at 07:31
  • @Prof.Hijibiji It does. There is a compact set $K$ contained in $E$ with positive measure and the projection of $K$ is a compact subset of the the projection of $E$ with positive measure. – Kavi Rama Murthy Mar 21 '19 at 07:35
  • Thanks once again to you sir. Thanks to inner regualirity property of Lebesgue measure also – Prof.Hijibiji Mar 21 '19 at 07:41
  • Just to say that the sentence "Projections need not be measurable" can be a bit misleading if taken out of context, as it means "the images of a set via a projection need not be measurable". The measurability of a projection as a map depends on the $\sigma$-algebra on the space, and this is usually required to make (at least) projections measurable, even in the example considered in the answer. – Gianmarco Mar 12 '25 at 15:31