Exercise measure theory: properties of a function Let $\Omega \subset R^n$ ($n\geq 2$) a bounded domain. Let $u \in L^{1}(\Omega)$.
For each number $c &gt;0$ define $A_c :=\{ x \in \Omega; u(x) &gt; c\}$. Define

$$ f(k) := \int_{A_k} (u(x)-k) dx$$.

I am reading  a book and the author says that

$$f(k) = \int_{k}^{+\infty} |A_t| dt  \ \ \ (*)$$

and 

$$ - f^{'}(k) = |A_k|\ \ \ \ (**)$$.

I have no idea of how to prove these two identities. Someone could help me ?

Thanks in advance.
That depends a bit on which definition of integrals you're using. I take the liberty to use the simplest for the purpose - sloppily expressed the area (or measure) of the region under the graph (and taking it negative if below the x-axis). 

The (*) should be rather straight forward using this definition as it's basically the same set we're measuring. You have to do the outer-inner measurement construct first to see this though.

The (**) follows from that we can use the estimate $f(k+h)-f(k)$ to be between $|h||A_k|$ and $|h||A_{k+h}|$.

If you're not using this definition it will be required to prove that $\int f(x)dx = |P|-|N|$ where $P = \{x, y: 0\le y\le f(x)\}$ and $N=\{x, y: f(x)\le y \le 0\}$.
That depends a bit on which definition of integrals you're using. I take the liberty to use the simplest for the purpose - sloppily expressed the area (or measure) of the region under the graph (and taking it negative if below the x-axis). 

The (*) should be rather straight forward using this definition as it's basically the same set we're measuring. You have to do the outer-inner measurement construct first to see this though.

The (**) follows from that we can use the estimate $f(k+h)-f(k)$ to be between $|h||A_k|$ and $|h||A_{k+h}|$.

If you're not using this definition it will be required to prove that $\int f(x)dx = |P|-|N|$ where $P = \{x, y: 0\le y\le f(x)\}$ and $N=\{x, y: f(x)\le y \le 0\}$.