Let $(X,\mathcal{F},\mu)$ be a $\sigma-$finite measure space. Let $\varphi_1,\varphi_2:X\to\mathbb{R}$ functions in $\mathcal{M}(X,\mathcal{F},\mathbb{R})$ such that $\varphi_1(x)\leq\varphi_2(x)$ for all $x\in X$. Consider the set $E=\{(x,t)\in X\times\mathbb{R}:\varphi_1(x)\leq t\leq\varphi_2(x)\}$ Let $F\in L^{1}(X\times\mathbb{R},\mathcal{F}\otimes\mathcal{B}(\mathbb{R}), \mu\otimes\lambda)$. Show that, $$E\in\mathcal{F}\otimes\mathcal{B}(\mathbb{R})$$ $$\int_{E}{F(x,t)}d\mu\otimes\lambda=\int_{X}{\int_{[\varphi_1(x),\varphi_2(x)]}{F(x,t)d\lambda(t)d\mu(x)}}$$
Where $\lambda$ is the lebesgue measure over $\mathcal{B}(\mathbb{R})$.
For the second question I use, Fubinni, but someone says to me that is wrong. Any help pls! Thanks!