I have been reading this excellent paper on Markov semigroups, in which the assertion is made that a markov semigroup $\mathcal{P: L^1 \longrightarrow L^1}$ is defined by $\frac {d\mu}{dm}$ for some expectation $\mu(\mathcal{A}) = \int_a ^b f(x)\mathcal{P}(x,\mathcal{A})m(dx)$ which I presume relates to the expectation of $f(x)$ over a given period $[a,b]$ or number of steps of the markov chain (for continuous and discrete time respectively).
I have little background in measure theory and I'm sure this doesn't help, however I'm struggling to see how this all comes together, and wonder if someone can help me out. I'm guessing that the m(dx) is referring to the way that the probability distribution $\mathcal{P}(x,\mathcal{A})$ changes over time?
I wonder if someone can give me a source (or even directly provide) some examples for calculating the expected value of a process using this method for a given $t$ and $x$? Something like a Poisson jump process would be particularly useful for my case.