I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both books talk about the original development of ergodic theory from statistical mechanics, saying that to any dynamical system you can associate a measure space, and the former proves the Poincare recurrence theorem and mentions that it was originally applied to celestial mechanics. Neither book mentions how to go about getting a measure space from a dynamical system, however. My question is then twofold:
How do you get a measure space from a dynamical system? I understand that the underlying space is the state space (or phase space?) of the system, but where does the measure come from?
What physical role does the measure on that space play, for instance, what does the measure being finite correspond to, or what does it being atomic correspond to?