Borel Measurable Set Related to Sections

Question

Let $E\subseteq\mathbb{R}^{n+1}$ be a Borel set and define \begin{equation} E^x=\{y\in\mathbb{R}\colon(x,y)\in\mathbb{R}^{n+1}\} \end{equation} for $x\in\mathbb{R}^n$, where we identify $\mathbb{R}^{n+1}$ with $\mathbb{R}^n\times\mathbb{R}$. Given an interval $I\subseteq\mathbb{R}$ define \begin{equation} X_I=\{x\in\mathbb{R}^n\colon0<\lambda(I\cap E^x)<\lambda(I)\} \end{equation} where $\lambda$ is the Lebesgue measure of $\mathbb{R}$. Is it necessarily true that $X_I$ is Borel measurable? If not, what further conditions can we impose to guarantee that $X_I$ is Borel measurable?

Answer 1

A part of the statement of Fubini's theorem asserts that for any Borel measurable set $E\in\mathcal{B}(\mathbb{R}^{n+1})$, $$ x\mapsto \lambda(E^x) $$ is a Borel measurable function. Note that $$ E^x \cap I = \left(E \cap [\mathbb{R}^{n}\times I] \right)^x. $$ Hence, it easily follows that $X_I$ is Borel measurable as a level set of a Borel measurable function.

It is more subtle if we consider a Lebesgue measurable $E$. We can show that $$ x\mapsto \lambda(E^x) $$ is defined for almost every $x\in\mathbb{R}^n$ and it admits a measurable version. Thus, your set $X_I$ is Lebesgue measurable in this case.