# 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.