Assume that $f:E \to [0,\infty]$ where $E \subseteq \mathbb{R}^n$ is a measurable set, and $f$ is $\mathbb{L}$-measurable. And use $x \in \mathbb{R}^n$ and $y \in \mathbb{R}$.
First I'm wondering why the subsets A and B stated below are measurable. $$A=\{(x,y) \in \mathbb{R}^{n+1} | 0 \le y < f(x), x \in E\}$$ $$B=\{(x,y) \in \mathbb{R}^{n+1} | 0 \le y \le f(x), x \in E\}$$
And then how can I conclude following equation for measure value?
\begin{align} \lambda(A)=\lambda(B)=\int_E f(x)dx & = \int_0^\infty \lambda(\{x \in E | f(x) >y\})dy \\ &= \int_0^\infty \lambda(\{x \in E | f(x) \ge y\})dy \end{align}