I'm trying to understand which probability inequality is used at the end of the following proof of Kingman's theorem in Revuz' Markov Chains book. He's considering a probability space with probability measure $m$ and a $m$-preserving (i.e. $m\circ\theta^{-1}=m$) measurable $\theta$.
It looks like Markov's inequality but this would yield $\le\varepsilon^{-1}\int\sup_{0\le r<k}s_k\:{\rm d}m\sum_{n\in\mathbb N}\frac1n$, where $\sum_{n\in\mathbb N}\frac1n=\infty$.
Recall that the Layer-Cake Representation will allow us to write $$\int \sup_{0 \leq r < k} s_r dm = \int_0^\infty m(\sup_{0 \leq r < k} s_r > t) dt$$ since $s_r$ is assumed to be positive.
Now by a change of variables, you get that \begin{align*} \int \sup_{0 \leq r < k} s_r dm =& \varepsilon \int_0^\infty m(\sup_{0 \leq r < k} s_r > \varepsilon s) ds \\ \geq & \varepsilon \int_0^\infty m(\sup_{0 \leq r < k} s_r > \varepsilon \operatorname{ceil}(s)) ds \\ =& \varepsilon \sum_{n=1}^\infty m(\sup_{0 \leq r < k} s_r > \varepsilon n) \end{align*} which rearranges to the desired inequality.