In this first chapter we shall describe some basic ideas in the ergodic theory of general measurable spaces. In later sections we shall specialize to diffeomorphisms of manifolds.

For the initiated we have included a more recent proof of the ergodic theorem, to help relieve the tedium. At the other extreme we have added an appendix (Appendix A) explaining some of the necessary background in measure theory.

Invariant measures.

In measure theory the basic objects are measurable spaces (X,ℬ), where X is a set and ℬ is a sigma algebra (or σ-algebra). In ergodic theory the basic objects are measurable spaces (X,ℬ) and a measurable transformation T: X→X (i.e. with T−1ℬ⊂ℬ). Given such a transformation T: X→X we want to consider those probability measures m: (ℬ→ℝ+ which are ‘appropriate’ for T in the following sense.

Definition. A probability measure m is called invariant (or more informatively, T-invariant) if m(T−1B) = m(B) for all sets B∈ℬ

This definition just tells us that the sets B and T−1 B always have the same ‘size’ relative to the invariant measure m. It is easy to see that if the transformation T is a bijection and its inverse T−1 : X→X is again measurable then the above definition is equivalent to asking that m(B) = m(TB) for all sets B∈ℬ

An alternative formulation of this definition is to ask that T*m=m where T*:ℳ→ℳ is the map on the set ℳ of all probability measures on X defined by (T*m)(B) = m(T−1B), for all B∈ℬ.