Reference for stochastic processes which helps moving from a basic level to a measure theory one

Question

I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains).

I've studied markov chains and measure theory but now I'm looking for something which helps me filling the gap and making this two topics converge.

I've already read: Measure, Integral and Probability - Marek Capinski, Peter E. Kopp

Maybe something with direct comparison (which writes the same probability both at a basic level and in measure theory) would be great!

Thanks in advance!

This post imported from StackExchange Physics at 2014-08-11 14:58 (UCT), posted by SE-user edwineveningfall