Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I'm really banging my head on this problem:

Let f be a real-valued measurable function on the measure space [tex](X,\mathcal{M},\mu).[/tex]

Define

[tex]\lambda_f(t)=\mu\{x:|f(x)|>t\}, t>0.[/tex]

Show that if [itex]\phi[/itex] is a nonnegative Borel function defined on [0,infinity), then

[tex]\int_0^{\infty}\phi(|f(x)|)d\mu=-\int_0^{\infty}\phi(t)d\lambda_f(t).[/tex]

A hint is given, which is to look at

[tex]\nu((a,b])=\lambda_f(b)-\lambda_f(a)=-\mu\{x:a<|f(x)|\leq b\},[/tex]

and argue that it extends uniquely to a Borel measure.

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

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