For some $\sigma$-finite measure space $(X,\mathcal{A},\mu)$ and a measurable function $f: X \to \mathbb{C}$, it is true that:
$$\int |f|^p = \int_0^\infty pt^{p-1} \mu(\{x: |f(x)|\geq t\})) dt$$
for positive, finite $p$.
I am pretty lost with this problem, and have several questions.
We can use the definition of the measure in terms of an integral to get
$$\int_0^\infty p t^{p-1} \mu(\{x:|f(x) \geq t\}) dt = \int_0^\infty pt^{p-1} \left[\int_X \chi_{\{x:|f(x) \geq t\}} (x) d\mu \right] dt$$
I assume the integral on the right-hand side is a Riemann integral, but can I justify doing the change of variables $u = t^{p-1}$ to obtain the following integral?
$$\int_0^\infty \mu(\{x:|f(x)|^p\geq u\}) du$$
My problem is that I have a Lebesgue integral as the integrand of (what I assume to be) the Riemann integral.
Then, how is this last integral in terms of $du$ equal to the $L_p$ norm?
Also, is it better to prove this for $f:X\to \mathbb{R}$ and then extend it to $\mathbb{C}$?
It is not assumed that$\mu$ is the Lebesgue measure (at least there's no reference to that in the question).