Let $E ⊆ R$ be a measurable set with |E|> 0. Prove that there exists $a ∈R, a\not= 0$, such that $|(E + a) ∩E|> 0$.

Question

Let $E ⊆ R$ be a measurable set with $|E|> 0$. Prove that there exists $a ∈R, a\not= 0$, such that $|(E + a) ∩E|> 0$.

To prove this statement, we will use the fact that Lebesgue measure is translation invariant, meaning that for any measurable set $E$ and any real number $a$, we have $|E+a| = |E|$. Using this property, we can show that if $|(E+a) ∩ E| = 0$ for all $a ∈ R$, then $|E| = 0$, which is a contradiction to our assumption that $|E| > 0$. I don't how to go further... Help please

Answer 1

Hint: $\int \int\chi_E(t-a)\chi_E(t)dtda>0$ by Fubini. Hence, there exists $a$ such that $ |(E+a)\cap E|=\int\chi_E(t-a)\chi_E(t)dt>0$.

Answer 2

Hint:

Consider the function $$f(x)= \int 1_E(y)1_{(x+E)}(y)dy=m(E\cap (x+E))$$

Then $f$ is continuous and $f(0) =m(E) >0$

Then there exists $N_{\delta}(0) $ such that $f>0 $ on $N_{\delta}(0) $