Why probability measures in ergodic theory?

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?

Solution 1:

The question isn't really about probability spaces, it's about finite measure. Usually the theory on classic ergodic theory (by classic I mean on finite measure spaces) is developed on probability spaces but it also works on any finite measure spaces, just take the measure normalized and everything will work fine. This hypothesis is really needed, some theorems doesn't really work on spaces that doesn't have finite measure, eg, Poincarré Recurrence Thm it's not true if you open this possibility. (Just take the transformation defined on the real line by $T(x)=x+1$. It is measure preserving but it's not recurrent.)

Specifically on the Birkhoff Thm, it still valid on $\sigma$-finite spaces but it doesn't give you much information about the limit. In fact, the Birkhoff' sum converges to 0.

But there's a nice theory going on $\sigma$-finite spaces with full measure infinity. Actually there is a nice book by Aaronson about infinite ergodic theory and some really good notes by Zweimüller. Things here change a bit, eg, you don't have the property given by Poincarré Recurrence (you have to ask it as a definition).Some of the results try to chance how you make the Birkhoff sum in order to get some additional information and can be applied to the calculus of Markov Chains. Another nice example that was object of recent study is the Boole's Transformation and it is defined by \begin{eqnarray*} B: \mathbb{R} &\rightarrow& \mathbb{R} \\ x &\mapsto& \dfrac{x^2-1}{x} \end{eqnarray*}

I don't know if I made myself very clear, but I recommend those texts. You should try it, it offers this theory and seek for the answer of your kind of question.

Aaronson, J. - An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs, AMS, 1997.

Zweimüller, R. - Surrey Notes on Infinite Ergodic Theory. You can get it here