real analysis - Finite positive measure

I currently don't find any approach to this question. In particular, how can I related a finite positive measure to a regular integration?

Suppose $\mu$ be a finite positive measure. Let $f$ be measurable, and $g$ is an increasing function in $C^1$. I am trying to prove the following property: $\int g(f)\,d\mu = \int_0^\infty \mu\{x:\ |f(x)|>t\} g'(t)\,dt$.

Answer

Just use Fubini:

$$\begin{align*} \int_0^\infty \mu\{x: |f(x)| > t\} g'(t) dt &= \int_0^\infty \int 1_{\{|f| > t\}}(x) \, g'(t) d\mu(x) dt \\ &= \int \int_0^\infty 1_{\{|f| > t\}}(x) \, g'(t) dt d\mu(x)\\ &= \int \int_0^{|f(x)|} g'(t) dt d\mu(x)\\ &= \int g(|f(x)|) - g(0) d\mu(x) \end{align*}$$

So if $g(0) = 0$ and you were searching for the property with $g(|f|)$ instead of $g(f)$ this is ok.