Lebesgue-Integral as Riemann-Integral This is a rather basic question on parts of the Lebesgue-Integral formula. Intuitively it should be clear, but I am searching for a mathematical explanation.
If $f: (\Omega, \mathcal{A}) \rightarrow (\mathbb{R}, \mathfrak{B}(\mathbb{R}))$ is a measurable function and $\mu$ a measure on $(\Omega, \mathcal{A})$ and $\mu(\Omega) < \infty$, than we can write the Integral of $f$ wrt $\mu$ as
$$\int f d\mu  = \int f^+ d\mu - \int f^- d\mu 
=\int_{0}^{\infty} \mu \{y|f(y) \geq x\} dx + \int_{-\infty}^{0} [\mu \{y|f(y) \geq x\} - \mu(\Omega)] dx.  $$
I don't get how to derive the second part, i.e. where does the expression $\mu \{y|f(y) \geq x\} - \mu(\Omega)$ come from? (I know that this concerns the potentially negative part of the function, but at the moment I am unable to get, how I derive this second expression). Some help would be awesome!

Hint:
$$ \mu( \{y \quad | \quad f(y) < x \}) + \mu (\{y \quad | \quad f(y) \ge x\} = \mu (\Omega)  $$
