# Projection of measurable set and Fσ set is measurable and Fσ set, respectively almost everywhere

lebesgue-measurereal-analysis

This is a proof from measure and integral by Richard and Antoni in page
90

Theorem: Let $$f(x,y)$$ be a measurable function on $$\mathbb{R}^{n+m}$$. Then for almost every $$x \in$$ $$\mathbb{R}^{n}$$, $$f(x,y)$$ is a measurable function of $$y \in\mathbb{R}^{n}$$. In particular, if $$E$$ is a measurable subset of $$\mathbb{R}^{n+m}$$, then the set $$E_x=$${$$y:(x,y) \in E$$} is a measurable in $$\mathbb{R}^{n}$$ for almost every $$x \in$$ $$\mathbb{R}^{n}$$

Proof:
First, if $$f$$ is a characteristic function $$\chi_E$$ of a measurable set $$E$$ in $$\mathbb{R}^{n+m}$$, then the two statements above are equal. In this case, write $$E=H$$ $$\cup$$ $$Z$$ ,where $$H$$ is of type $$F_{\sigma}$$ (ie: the union of closed set) in $$\mathbb{R}^{n}$$ and $$Z$$ is measure $$0$$,

Claim 1: $$E_x$$=$$H_x$$ $$\cup$$ $$Z_x$$, where $$H_x$$ is of type $$F_{\sigma}$$ and for almost every $$x$$, $$Z_x$$ is measure $$0$$
suppose the claim hold, then the result follow in this case.

Claim2:
If $$f$$ is any measurable function on $$\mathbb{R}^{n+m}$$, consider the set $$E(a)$$={$$(x,y):f(x,y)>a$$}, clearly $$E(a)$$ is a measurable in $$\mathbb{R}^{n+m}$$, show that the set $$E(a)_x$$= {$$y:(x,y) \in{E(a)}$$} is a measurable set in $$\mathbb{R}^{n}$$ for almost every $$x$$ $$\in$$ $$\mathbb{R}^{n}$$

For claim $$1$$: that is to say the projection of a
$$F_{\sigma}$$ is also$$F_{\sigma}$$, I have a question that why it is closed, (ie: the projection of the closed set is not general true), but why this hold?

For claim two: it says that the section of a measurable set is measurable, but I have no idea to prove this fact

can someone help me, thanks

For claim 1, it is enough to show that the projection of {$$x$$}$$\times B$$ is a closed set in ambient space if and only if the projection is also be closed set. This proof is easy, since the projection is an open mapping, so the result follows.