Let $E \subseteq \mathbb{R^n}$ a measurable set, such as for almost every $x \in \mathbb{R^n}$ we have $|E \triangle (E+x)|=0$ (Where $\triangle$ means simetric difference between two sets and $|\cdots|$ is the Lebesgue measure). Then $|E|=0$ or $|E^c|=0$.
Well I'm really stuck here. I would like to apply Fubini-Tonelli's theorem to the function $\chi_{E\triangle (E+x)}$ (which is positive and measurable). But first I need to calculate the section of the set $E \triangle (E+x)$ with respect to the variable $x_{1}$ (for example). Where $(E \triangle (E+x))_{x_1}=\{(x_{2},\cdots,x_{n}) \in \mathbb{R^{n-1}} : (x_{1}, \cdots, x_{n}) \in E \triangle (E+x)\}$ I don't know how to do it. Thanks a lot for your help.
You don't need the sections. Let $h(x) = \lvert E\triangle (E+x)\rvert$. Then you know
$$\begin{align} 0 &= \int_{\mathbb{R}^n} h(x)\,dx\\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \chi_{E\triangle (E+x)}(y)\,dy\,dx\\ &= \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \chi_E(y) + \chi_E(y-x) - 2\chi_E(y)\chi_E(y-x)\,dy\,dx. \end{align}$$
The integrand is measurable and non-negative, so you can exchange the order of integration.