I was reading the proof of the Portmanteau Theorem, when this statement was made:
$\int{fdP} = \int_{0}^{supf}{P\{{f\geq t\}}dt}$
Here the P is a probability measure on a metric space, and $f\ge0$ is a real-valued continuous bounded function on the same space. Now I am familiar with measure theory and intuitively this makes sense to me. But for some reason I cannot justify it on paper. Any help would be appreciated as to why this is so.
