Definable equivalence relations on Polish spaces

Citation

Ditzen, Achim (1992) Definable equivalence relations on Polish spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/rf5s-m868. https://resolver.caltech.edu/CaltechTHESIS:11232011-134429806

Abstract

In the first part of this work we deal with the classification of definable equivalence relations on Polish spaces, where we take definable to mean inside some model of determinacy: We work in ZF+DC+AD_R. The classification is up to bireducebility (denoted by E~F), that is if E and F are equivalence relations on the Baire space N, then E ~F, if there is a mapping f : N → N with ∀x, y ∈ N (xEy ⇔ f(x)Ff(y)), called a reduction of E into F, and vice versa.

As two equivalence relations on Polish spaces are bireducible just in case there is a bijection between their quotient spaces, our results apply to de-finable cardinality theory, too. We show that up to bireducibility there are only four infinite hypersmooth equivalence relations: equality on the integers, equality on the Baire space, E_o on the Cantor space 2^ω given by αE_oβ ⇔ ∃n ∈ w∀m > n (a(m) = ,β(m)), and E_1 on the countable product of Cantor space (2^ω)^ω given by αE_0^β ⇔ ∃n ∈ w∀m > n (a_m = β_m).

Even though we only develop the theory for the context of ADR, it is clear from the proofs that our results apply to a variety of other settings, such as the one encountered in the second part.

In the second part of the thesis we deal with countable Borel equivalence relations E on Polish spaces X, that is with equivalence relations which have countable classes and Borel graphs. The space M of probability measures on these spaces is again Polish. Of special interest are invariant measures (i.e. those which are preserved under bijections f : X → X with f(x)Ex, so called automorphisms), quasiinvariant measures (i.e. those whose measure class is preserved under automorphisms), and ergodic measures (i.e. those which assign full or null measure to E-invariant Borel sets).

We show that the collections of ergodic measures and of ergodic quasiin-variant measures are Borel. We also classify the complexity of the σ-ideal of nullsets with respect to all invariant measures, showing that this ideal is П^1_1 in the codes of ∆^l_1 and ∑^l_1 sets, and that the σ-ideal of compact nullsets with respect to all invariant measures is П^0_2 if the collection of invariant ergodic measures is at most countable, and П^1_1 -complete otherwise.

Thesis Files