6-2

Let F(x,y)=f(x) and G(x,y)=g(y) for all x,y\in\mathbb{R}^n
. Note that
\{(x,y)\in\mathbb{R}^n\times\mathbb{R}^n|F(x,y)>\alpha\}=\{x\in\mathbb{R}^n|f(x)>\alpha\}\times\mathbb{R}^n
is measurable by Lemma 5.2. Same reason can apply to G(x,y). So both F(x,y) \;\text{and}\;G(x,y) are measurable in \mathbb{R}^n \times \mathbb{R}^n. Hence F(x,y)G(x,y)=f(x)g(y) is measurable in \mathbb{R}^n \times \mathbb{R}^n.
Let E_1 \; \text{and}\; E_2 be measurable subsets of \mathbb{R}^n. Then \chi_{E_1} and \chi_{E_2} are measurable in \mathbb{R}^n. Then by what we proof above, \chi_{E_1}(x)\chi_{E_2}(y) is measurable in \mathbb{R}^n \times \mathbb{R}^n. Since \chi_{E_1}(x)\chi_{E_2}(y)=\chi_{E_1\times E_2}(x,y), so E_1 \times E_2 is measurable in \mathbb{R}^n \times \mathbb{R}^n.
Lastly, we have
\begin{align}
|E_1\times E_2|&=\iint\limits_{\mathbb R^n\times\mathbb R^n} \chi_{E_1\times E_2}(x,y)dxdy
\\&=\iint\limits_{\mathbb R^n\times\mathbb R^n} \chi_{E_1}\chi_{E_2}(x,y)dxdy
\\&=\int_{\mathbb R^n}\chi_{E_1}(x)dx\int_{\mathbb R^n}\chi_{E_2}(y)dy
\\&=|E_1||E_2|.
\end{align}