Integration using a cumulative distribution function

1. Jul 29, 2011

laonious

Hi all,
I'm really banging my head on this problem:
Let f be a real-valued measurable function on the measure space $$(X,\mathcal{M},\mu).$$
Define
$$\lambda_f(t)=\mu\{x:|f(x)|>t\}, t>0.$$
Show that if $\phi$ is a nonnegative Borel function defined on [0,infinity), then
$$\int_0^{\infty}\phi(|f(x)|)d\mu=-\int_0^{\infty}\phi(t)d\lambda_f(t).$$

A hint is given, which is to look at
$$\nu((a,b])=\lambda_f(b)-\lambda_f(a)=-\mu\{x:a<|f(x)|\leq b\},$$
and argue that it extends uniquely to a Borel measure.

This is straightforward, I think, as closed intervals form a semi-ring and $\nu$ is a premeasure. I'm just not sure where to go from here. Any help would be greatly appreciated, thanks!

2. Jul 30, 2011

mathman

It looks like it could be solved by using the elementary approach to defining Lebesgue integrals - dividing the y (or f) axis into strips and taking a limit as the strip widths go to 0.

The strips would correspond to (a,b] in the hint.