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This is a question from Stromberg related to Steinhaus' Theorem:

If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval.

I can't quite see how to modify the Steinhaus proof though.

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possible duplicate: math.stackexchange.com/questions/59769/theorem-of-steinhaus –  t.b. Dec 1 '11 at 7:28

1 Answer 1

up vote 23 down vote accepted

I don't know what proof of Steinhaus theorem is used, but we can show the following result:

If $A$ and $B$ have a positive Lebesgue measure, then $A+B$ contains an interval.

We can assume that $A$ and $B$ have finite measure. Indeed, if $\lambda(A)$ is infinite, $A=\bigcup_{n\in\mathbb N}A\cap\left[-n,n\right]$ and we only have to pick $n_0$ such that $\lambda(A\cap \left[-n_0,n_0\right])>0$. If $n_1$ is such that $\lambda(B\cap \left[-n_1,n_1\right])>0$, and we have shown the result for $A$ and $B$ of finite measure, then $A+B\supset (A\cap \left[-n_0,n_0\right])+(B\cap \left[-n_1,n_1\right])\supset I$ and we are done.

Thank to the fact the indicator functions are in $L^2$ and the density of the continuous functions with compact support in $L^2(\mathbb R)$ $$f\colon x\mapsto \mathbf{1}_A*\mathbf{1}_B(x)=\int_{\mathbb R}\mathbf{1}_A(x-t)\mathbf {1}_B(t)d\lambda(t)$$ is continuous . Hence the set $O:=\left\{x\in\mathbb R,f(x)>0\right\}$ is open. Since $\int_{\mathbb R}f(x)d\lambda(x)=\lambda(A)\cdot\lambda(B)>0$, $O$ is non-empty and therefore contains an open non-empty interval $I$. If $x\notin A+B$, $A\cap(-B+x)=\emptyset$. Indeed, if $y\in A\cap(-B+x)$ then $y=a$ for some $a\in A$, and $y=-b+x$ for some $b\in B$, hence $x=a+b$. So if $x\notin A+B$, $f(x)=0$, and taking the complement, if $f(x)\neq 0$ then $x\in A+B$, hence we got $$I\subset O\subset A+B.$$

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extremely clear. Thanks. –  Bombyx mori Jul 28 '12 at 7:04

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