Definition For a discrete group $\Gamma$, we let Prob$(\Gamma)$ be the space of all probability measures on $\Gamma$: $$Prob(\Gamma)=\{\mu\in l^{1}(\Gamma): \mu\geq0,~\sum\limits_{t\in\Gamma}\mu(t)=1\}.$$
Now we set $F(\mu, r)=\{t\in \Gamma:~\mu(t)>r\}$. Can we conclude that $\int_0 ^1 |F(\mu,~r)|dr=1$? (Here, $|.|$ is the cardinal number of set.)
For a statement $P$ let $[P]$ denote $1$ if $P$ is true and $0$ if it's false. We have:
$$ \int_0^1 |F(\mu,r)|\;dr =\int_0^1 \sum_{t\in \Gamma} [\mu(t)>r]\;dr = \sum_{t\in\Gamma} \int_0^1 [\mu(t)>r] \;dr = \sum_{t\in\Gamma} \mu(t) = 1. $$