I just had a look at Walters' introductory book on ergodic theory and was struck that the book always sticks to probability measures. Why is it the case that ergodic theory mainly considers probability measures? Is it that the important theorems, for example Birkhoff's ergodic theorem, is true only for probability measures? Or is it because of the relation with concepts from thermodynamics such as entropy?
I also wish to ask one more doubt; this one slightly more technical. Probability theory always works with the Borel sigma algebra; it is rarely the case that the sigma algebra is enlarged to the Lebesgue sigma algebra for the case of the real numbers(for defining random variables) or the unit circle, for instance. In ergodic theory, do we go by this restriction, or not? That is, when ignoring sets of measure zero, do we have that subsets of measure zero are measurable?