Dynamical Systems Seminar

The talk is devoted to the problem of ergodic decomposition for measures quasi-invariant under actions of inductively compact groups, for instance, the infinite symmetric or the infinite unitary group.

As a simple case, let us first consider a continuous transformation of a compact metric space. The space of probability measures invariant under the transformation is itself a compact metric space, and its extremal points are exactly ergodic probability measures. Choquet's Theorem now allows one to represent a given invariant probability measure as an integral over the space of ergodic probability measures. Such a representation is called an ergodic decomposition. For actions of locally compact groups with a quasi-invariant measure, ergodic decompositions using Choquet's Theorem have been constructed by Greschonig and Schmidt. Note, however, that inductively compact groups are not locally compact.

Rohlin proposed a different way of constructing ergodic decompositions using his theory of measurable partitions. The advantage of Rohlin's construction is that it does not make any topological assumptions and can be applied to any measurable automorphism of a Lebesgue probability space. On the other hand, Rohlin's construction, unlike a Choquet-type construction, essentially relies on the ergodic theorem.

In the talk, Rohlin's approach will be used to establish existence and uniqueness of ergodic decompositions for measurable actions of inductively compact groups with a quasi-invariant measure. Instead of the ergodic theorem, the martingale convergence theorem is used.