The definition $E(x)=\left\{y\big|y\in\mathbb{R}^q\land <x,y>\in E\right\}$ is all the points that in E while $x$ is selected.
And furthermore, $E$'s projection in $\mathbb{R}^p$ is also an measurable set in $\mathbb{R}^p$.
In fubini it is said that if $E$ is measurable in $\mathbb{R}^{p+q}$, then for almost every where $x\in\mathbb{R}^p$ that $E(x)$ is measurable can be derived.
So I would further like to know if the reverse is true??
And if it is, may I have the prove, thanks a lot!