Property of Lebesgue measure in $mathbb{R}^2$, part 2

Question

Let $I=Atimes B,$ where $A,Bsubset mathbb{R}$ are closed sets of positive Lebesgue measure, and $Esubset mathbb{R}^2,$ be a set of zero Lebesgue measure. Is it true that $$overline{Isetminus E}=I?$$
When $I=[0,1]times [0,1]$ the answer is true and one can find a solution in the following link.
Property of Lebesgue measure in $mathbb{R}^2$

halrankard · Answer

No. Set $A=B=[0,1]cup{2}$. Let $E$ be the singleton ${(2,2)}$. Then
$overline{Isetminus E}neq I$.

Property of Lebesgue measure in $mathbb{R}^2$, part 2

One Answer

Add your own answers!

Ask a Question