I assume that you are asking for the construction of the probability space for a given LMP. The definition of the probability space becomes more involved once you allow continuous state spaces.

The usual way (in my oppinion) would be to construct the probability spaces via Borel $\sigma$-algebras. The following book is a good reference for such probability spaces: Robert B. Ash and Catherine A. Doleans-Dade. Probability & Measure Theory. Elsevier Science, 2000.

If that should not suffice, e.g. if you need a construction providing Lebesgue measurability, you could take a look at this paper, which is on a related system model (Continuous-time markov decision processes): http://link.springer.com/article/10.1007%2Fs00236-011-0140-0

I assume that you are asking for the construction of the probability space for a given LMP. Although, I do not have a particular reference for this construction, there are a few closely related constructions that might help you.

The usual way (in my oppinion) would be to construct the probability spaces via Borel $\sigma$-algebras. The following book is a good reference for such probability spaces: Robert B. Ash and Catherine A. Doleans-Dade. Probability & Measure Theory. Elsevier Science, 2000.

If that should not suffice, e.g. if you need a construction providing Lebesgue measurability, you could take a look at this paper, which is on a related system model (Continuous-time markov decision processes): http://link.springer.com/article/10.1007%2Fs00236-011-0140-0