The problem is the following
Let $E\subseteq\mathbb{R}$ be a Lebesgue measurable set such that $\lambda(E\Delta (E+x))=0$ for every $x$ in a dense subset of $\mathbb{R}$. Then either $\lambda(E)=0$ or $\lambda(E^{c})=0$.
What I have done is the following:
$\begin{align*} 0 &=\lambda(E\Delta (E+x))\\ &=\lambda(E)+\lambda(E+x)-2\lambda(E\cap (E+x))\\ &=2\left(\lambda(E)-\lambda(E\cap (E+x))\right) \end{align*}$
where the second equality follows from inclusion-exclusion for symmetric difference and the third by the invariance of Lebesgue measure. Then,$$\lambda(E)=\lambda(E\cap (E+x))$$ where the last identity holds for all $x$ in the dense subset.
It can be shown that $\lambda(E\cap (E+x))$ is a continuous function, so that the identity holds for all real $x$. If $E$ happens to be bounded the result clearly follows; but I'm unable to prove it in the general case.
This question has already answers here and here, but I would like to conclude from the work I have already done. The natural way to proceed is to suppose $\lambda(E)>0$ and conclude $\lambda(E^c)=0$ or viceversa, but I'm stuck.
