Borel set

Abstract: Let p be any finite positive measure on (the Borel sets of) a complete separable metric space X. We shall say that a random probability measure P* on X has a Ferguson distribution with parameter p if for every finite partition (B1, * . *, B) of X the vector p*(B,), * * *, p*(B,) has a Dirichlet distribution with parameter (Bj), *--, cp(B,) (when p(B,) = 0, this means p*(B,) = 0 with probability 1). Ferguson (3) has shown that, for any p, Ferguson p* exist and when used as prior distributions yield Bayesian counterparts to well-known classical nonpa- rametric tests. He also shows that p* is a.s. discrete. His approach involves a rather deep study of the gamma process. One of us (1) has given a different and perhaps simpler proof that Ferguson priors concentrate on discrete distributions. In this note we give still a third approach to Ferguson distributions, exploiting their connection with generalized Polya urn schemes. We shall say that a sequence (X,, n > 1} of random variables with values in X is a Poilya sequence with parameter 1a if for every B c X (1) P(X1 e B) = p(B)/p(X) and (2) P{X,+1 e B I1 **,, X = pn(B)/1p(X) where p. = p + 3 l(Xi) and 3(x) denotes the unit measure concentrating at x. Note that, for finite X, the sequence {XJ} represents the results of successive draws from an urn where initially the urn has p(x) balls of color x and, after each draw, the ball drawn is replaced and another ball of its same color is added to the urn. Note also that, without the restriction to finite X, for any (Borel measurable) function zS on X, the sequence {0(X")} is a P6lya sequence with parameter qSp, where q4(A) = p{l e Al. We now describe the connections between Polya sequences and Ferguson distributions.

Abstract: ON THE TRANSLOCATION OF MASSES L. V. Kantorovich∗ The original paper was published in Dokl. Akad. Nauk SSSR, 37, No. 7–8, 227–229 (1942). We assume that R is a compact metric space, though some of the definitions and results given below can be formulated for more general spaces. Let Φ(e) be a mass distribution, i.e., a set function such that: (1) it is defined for Borel sets, (2) it is nonnegative: Φ(e) ≥ 0, (3) it is absolutely additive: if e = e1 + e2+ · · · ; ei∩ ek = 0 (i = k), then Φ(e) = Φ(e1)+ Φ(e2) + · · · . Let Φ′(e′) be another mass distribution such that Φ(R) = Φ′(R). By definition, a translocation of masses is a function Ψ(e, e′) defined for pairs of (B)-sets e, e′ ∈ R such that: (1) it is nonnegative and absolutely additive with respect to each of its arguments, (2) Ψ(e, R) = Φ(e), Ψ(R, e′) = Φ′(e′). Let r(x, y) be a known continuous nonnegative function representing the work required to move a unit mass from x to y. We define the work required for the translocation of two given mass distributions as W (Ψ,Φ,Φ′) = ∫

Abstract: 1 Fundamentals- 11 Operators and C*-algebras- 12 Two density theorems- 13 Ideals, quotients, and representations- 14 C*-algebras of compact operators- 15 CCR and GCR algebras- 16 States and the GNS construction- 17 The existence of representations- 18 Order and approximate units- 2 Multiplicity Theory- 21 From type I to multiplicity-free- 22 Commutative C*-algebras and normal operators- 23 An application: type I von Neumann algebras- 24 GCR algebras are type I- 3 Borel Structures- 31 Polish spaces- 32 Borel sets and analytic sets- 33 Borel spaces- 34 Cross sections- 4 From Commutative Algebras to GCR Algebras- 41 The spectrum of a C*-algebra- 42 Decomposable operator algebras- 43 Representations of GCR algebras

Abstract: Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets Coanalytic sets and pi 1/1 sets The Gandy-Harrington topology Polish Groups Metrics on topological groups Polish groups Continuity of homomorphisms The permutation group S Universal Polish groups The Graev metric groups Polish Group Actions Polish G-spaces The Vaught transforms Borel G-spaces Orbit equivalence relations Extensions of Polish group actions The logic actions Finer Polish Topologies Strong Choquet spaces Change of topology Finer topologies on Polish G-spaces Topological realization of Borel G-spaces Theory of Equivalence Relations Borel Reducibility Borel reductions Faithful Borel reductions Perfect set theorems for equivalence relations Smooth equivalence relations The Glimm-Effros Dichotomy The equivalence relation E0 Orbit equivalence relations embedding E0 The Harrington-Kechris-Louveau theorem Consequences of the Glimm-Effros dichotomy Actions of cli Polish groups Countable Borel Equivalence Relations Generalities of countable Borel equivalence relations Hyperfinite equivalence relations Universal countable Borel equivalence relations Amenable groups and amenable equivalence relations Actions of locally compact Polish groups Borel Equivalence Relations Hypersmooth equivalence relations Borel orbit equivalence relations A jump operator for Borel equivalence relations Examples of Fsigma equivalence relations Examples of pi 0/3 equivalence relations Analytic Equivalence Relations The Burgess trichotomy theorem Definable reductions among analytic equivalence relations Actions of standard Borel groups Wild Polish groups The topological Vaught conjecture Turbulent Actions of Polish Groups Homomorphisms and generic ergodicity Local orbits of Polish group actions Turbulent and generically turbulent actions The Hjorth turbulence theorem Examples of turbulence Orbit equivalence relations and E1 Countable Model Theory Polish Topologies of Infinitary Logic A review of first-order logic Model theory of infinitary logic Invariant Borel classes of countable models Polish topologies generated by countable fragments Atomic models and Gdelta-orbits The Scott Analysis Elements of the Scott analysis Borel approximations of isomorphism relations The Scott rank and computable ordinals A topological variation of the Scott analysis Sharp analysis of S -orbits Natural Classes of Countable Models Countable graphs Countable trees Countable linear orderings Countable groups Applications to Classification Problems Classification by Example: Polish Metric Spaces Standard Borel structures on hyperspaces Classification versus nonclassification Measurement of complexity Classification notions Summary of Benchmark Equivalence Relations Classification problems up to essential countability A roadmap of Borel equivalence relations Orbit equivalence relations General SIGMA 1/1 equivalence relations Beyond analyticity Appendix: Proofs about the Gandy-Harrington Topology The Gandy basis theorem The Gandy-Harrington topology on Xlow References Index