Since two equivalence relations on field spaces are bireducible only if there is a bijection between their quotient spaces, our results also apply to definable cardinality theory. In the second part of the thesis, we consider numerical Borel equivalence relations E on field spaces X, i.e. equivalence relations that have numerical classes and Borel graphs. In this chapter, we prove two dichotomy theorems about reducibility and embeddability relations of equivalence relations.
We are interested here in these notions for "definable" objects on Polish spaces, that is, where the spaces are Polish spaces and the equivalence relations and maps are "definable". On the other hand, Eo is the increasing union of equivalence relations with finite classes, namely those equivalence relations on 2W that relate elements that match after a fixed point k E w. We call equivalence relations with finite equivalence classes finite, and the increasing union of finite equivalence relations hyperfinite.
Since E1 = Uk Fk, we have E = Uk Fk and both are increasing unions of smooth equivalence relations. We will show (lemma 6) that such E are hyperfinite, i.e. increasing union of finite equivalence relations.
Auxiliary Pointclasses
The only linear sequences (up to isomorphism) that satisfy this property are finite sequences, Z, w, and w*, the inverse of w. If [X]E has order type w, let y, Z E [XJE yRz if y, z both have an even number of R' predecessors and yR' z or y, z both have an odd number of R' predecessors and zR' y or y has an odd number of R' predecessors and z has an even number of R' predecessors. Put xRn+1y iff xRnY or xEn+1y and not xEnY, and Rn-smallest element in [X]En is lexicographically smaller than Rn-smallest element in [yJEn.
Trees And Tree Structures
Here 11"0 and 11"1 are the projections on the first and second coordinates of the product space, respectively. Let To and Tl be the two subtrees of T that remain when e is removed from E(T). Let To and Tl be the two subtrees of T that remain when e is removed from E(T).
The First Dichotomy Theorem
- The Embedding
- The Game G
- Trees on 2 n
- Construction of the binary tree
- Stage 0
- Stage 1
- Stage n + 1
Since A~ReA~ we have A~ReA~ for all u, v E VeT). vVe will construct the continuous embedding of f of Eo in EIX by constructing a sequence {i : nEw} of positive integers and a perfect binary tree {A~ : i ::; i, S E 2n, nEw} of f-subsets of X such that. For k = 0 there is nothing to show. of the embedding lemma we will play the following game:. For this we should fix a prewell ordering of the reals of length A and should encode the ordinals with reals of the appropriate rank relative to this prewell ordering.
It's easy to modify the following argument to work with the encrypted version of the game. Using Lemma 13 and the tree T proving that E is A-Suslin, we have Lemma 15. Player II has a winning strategy in G'. Before we can give the construction of the complete binary tree of r-subsets { A ~} of X, we will need to construct a directed tree Tn on each 2n.
We tried to reduce the A~ to make sure (A), that is, that the closures of the A~n+J are pairwise disjoint. To guarantee (C), successively use lemma 10 on each of the vertices of Tn+1.
The Second Dichotomy Theorem
Since A' = Un(A' n nm>n Yn,m), the effective reduction property allows us to find a uniform r set {An : nEw} such that A' = Un An. Also note that F~ is uniformly smooth; that is, we have uniform r nt recursive reductions ¢n from F~ to equality at 2W.
To make sure (b), it is convenient to consider the following game G and use the fact that A is A-Suslin to show that II has a winning strategy. Alternatively, one could use the argument from the proof below directly in the construction. If (T, IT) is a labeled tree, we say that vertices s, t E V(T) are n-linked if there is an edge labeled n between them, and we denote this by s.ILt.
If all edges in the path between s and t have labels ~n, we say sand tare is n-jointed and write s-n_t. Using the fact that M : w -> w will be strictly increasing and that the Fn equivalence relations increase, we can rewrite (c) as. Proof: We prove the statement of the lemma for subtrees T of Tp by induction on the number of n-equivalence classes of V(T).
We assume without loss of generality that Ck is an end node of T' and that Ck-l is m-connected to Ck.
A Perfect Set Theorem for n-ary Rela- tions
Proof of the Perfect Set Theorem
Since Ii (J) followed, we have Vi R( ai, f3i, Ii) and we get a contradiction with the premises of the theorem. .Af2 x Cl) Here, a solid arrow indicates that the string at the top of the arrow has been copied and is the same as the string at the tail of the arrow.
A solid arrow with 1/2 inset in the center indicates that the set at the head is a non-empty subset of the set at the tail of at most half the diameter. A broken arrow indicates that the set at the head is a projection of the set at the tail. The filled dots indicate that the sets are obtained from the previous ones by shrinking; the others have just been copied.
This completes the construction of the A~ and the B~ and thus the proof of the theorem.
Chapter 2
Measures for Countable Borel Equivalence Relations
Ergodic Measures for Countable Borel Equivalence Relations
- Uniformities
It can be easily shown that these notions are independent of G; see e.g. Dougherty-Jackson-Kechris [aJ.) A measure f-l is called E-ergodic if f-l(A) = 0 or 1, for any Borel E-invariant (that is, A is the union of E-equivalence classes) set A ~ X. Dougherty-Jackson-Kechris (a]) that the sets I and £I of E-invariant and E-ergodic E-invariant measures, respectively, are Borel in M(X), and so is Q, the set of E-quasi-invariant measures. Then the set £ of ergodic probability measures is Borel in the space of probability measures M on X.
This improves on the result of Krieger [71, p.lS7], who calculates that the set of almost invariant, ergodic probability measures is ill. Then the set of ergodic probability measures £ is Borel on the space of probability measures M on X Now the mapping f-l -+ ff-l is Borel and reduces the set of ergodic measures of E to the set of ergodic measures of ElY.
First, we note that the set £ of ergodic probability measures can be reduced to a set of quasi-invariant, ergodic measures. Thus, for every E £ is Borel reducible to £Q, so it suffices to show that £Q is Borel. Let QP be the Borel set of all quasi-invariant measures f-l on X such that for all 9 E G,.
Let us mention that, for specific examples, a more careful analysis of the proof yields concrete upper bounds for the Borel complexity of the space of ergodic measures. Since any two uncountable, standard Borel spaces X and Yare are Borel isomorphic via a Borel measurable bijection f, we can assume in the rest of the paper that we are dealing with perfect Polish spaces. An important Polish space is the Baire space N of all functions from w to w, with the product topology, taking w to have the discrete topology.
In a Polish space X is a set :E~ if it is the image of a closed subset of a Polish space under a continuous function. It is well known that a subset of a Polish space is Borel if :Et and II~. This gives rise to a standard way of encoding Borel subsets in a Polish space X: We can find II~ sets C ~ Nand D, b ~ N X X like this.
2.1.2- Proof of Theorem 2
Proof of the Effective Ergodic Decomposition Theorem
Iljlloo ~ Ilflloo,
- Ideals of Compressible Sets
- An Algebra of .6.i-Sets
- Proof of the Key Lemma
By the theorem of Burkholder-Chow about iterates of conditional-expectation operators mentioned in the previous section, . for every quasi-invariant measure J1 E QP. By setting j = 0 on R, we can and do assume that the above equality applies everywhere; that is, we can assume that cp is Q-linear on W . We can then transform J-lx onto an E-invariant QP null set J-l, to ensure that J-lx E QP for all x E X.
There is a ~t(a) compression of [AlE. Thus we find a ~Ha) compaction for F, which we can transfer via g to a ~Ha) compaction of [AlE. Recall that the collection K(X) of compact subsets of a compact metric space X is again a compact metric space with the Hausdorff topology, i.e. the topology generated by shape sets. An O" ideal is checked if there is an Et in the codes of rrg-set collection A of rrg-sets compatible with I and with 0 E A.
By Feldman-Moore [77] we can assume that we have a countable group G acting on X in a .6.~ fashion, so that E is the orbital equivalence relation of that G action. So we can transfer this structure by the projection on X, which is a bijective from G to X. Proof: Let for each i (Ci, di, Td be given by the previous fact for Ci.
Prool: Since ~~ sets are uniformly closed under boolean operations and take models under ~i functions, there is a recursive in the ~i codes. Suppose further that a code for 1 can be found recursively in the codes for A and B. is a compression of AC, and on the remainder. Furthermore, codes for all these objects can be found recursively in the codes for A and B.
Recall that Gn was a graph of gn, where gn was the nth function in the numbering of the group that induces E, such that go was the identity map. The following mapping c is a compression of AQmnQ~: For x E AQmnQ~, suppose that x is the kth element (in lexicographic order) in the root f (x) under f. Since the set of points where the limit does not exist is at most a set where Lemma 12 does not hold or where the sequence of reference sets does not have the required properties, we can recursively find the code for this set and for its compression from the code A and the sequence of reference sets.
So almost everywhere on Pn we have mX(Fn) < mX(A) and so we can apply the following lemma.
Bibliography