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.
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