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 byαE_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.
