Ideals without ccc

Given Polish spaces X and Y and a Borel set S ⊆ X × Y with countable sections, we describe the circumstances under which a Borel function f : S → R is of the form f(x, y) = u(x) + v(y), where u : X → R and v : Y → R are Borel. This turns out to be a special case of the problem of determining whether a real-valued Borel cocycle on a countable Borel equivalence relation is a coboundary. We use several Glimm-Effros style dichotomies to give a solution to this problem in terms of certain σ-finite measures on the underlying space. The main new technical ingredient is a characterization of the existence of type III measures of a given cocycle. Suppose that S ⊆ X×Y and G is a group. A coordinatewise decomposition of a function f : S → G is a pair (u, v), where u : X → G, v : Y → G, and ∀(x, y) ∈ S (f(x, y) = u(x)v(y)). If X and Y are Polish spaces, G is a standard Borel group, and u and v are Borel, then we say that (u, v) is a Borel coordinatewise decomposition of f . Our main goal here is to show that when S is a Borel set with countable sections, f : S → G is Borel, and G = 〈R,+〉, the existence of a Borel coordinatewise decomposition can be characterized in terms of certain σ-finite measures on the disjoint union of X and Y (by a measure on a Polish space, we shall mean always a measure on its Borel subsets). Before getting to this, however, we consider first the existence of coordinatewise decompositions, without imposing any definability restrictions. ∗The author was supported in part by NSF VIGRE Grant DMS-0502315.

We define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

Define to be the smallest cardinality of a function f: X→Y with I, X, Y, ⊆ 2ω such that there is no Borel function g ⊇ f. In this paper we prove that it is relatively consistent with ZFC to have b < where b is, as usual, smallest cardinality of an unbounded family in Ωω. This answers a question raised by Zapletal.We also show that it is relatively consistent with ZFC that there exists X ⊆ 2ω such that the Borei order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin.

We establish dichotomy results concerning the structure of Baire class 1 functions. We consider decompositions of Baire class 1 functions into continuous functions and into continuous functions with closed domains. Dichotomy results for both of them are proved: a Baire class 1 function decomposes into countably many countinuous functions, or else contains a function which turns out to be as complicated with respect to the decomposition as any other Baire class 1 function; similarly for decompositions into continuous functions with closed domains. These results strengthen a theorem of Jayne and Rogers and answer some questions of Steprāns. Their proofs use effective descriptive set theory as well as infinite Borel games on the integers. An important role in the proofs is played by what we call, in analogy with being Wadge complete, complete semicontinuous functions. As another application of our study of complete semicontinuous functions, we generalize some recent theorems of Jackson and Mauldin, and van Mill and Pol concerning measures viewed as examples of complicated semicontinuous functions. We also prove that a Borel set A A is either Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } or there is a continuous injection ϕ : ω ω → A \phi :\; \omega ^{\omega }\to A such that for any Σ α 0 \boldsymbol \Sigma ^{0}_{\alpha } set B ⊂ A B\subset A , ϕ − 1 ( B ) \phi ^{-1}(B) is meager. We show analogous results for Borel functions. These theorems give a new proof of a result of Stern, strengthen some results of Laczkovich, and improve the estimates for cardinal coefficients studied by Cichoń, Morayne, Pawlikowski, and the author.

A subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually calledPolish spaces. Typical examples are ℝn, ℂn, (separable) Hilbert space and more generally all separable Banach spaces, theCantor space2ℕ, theBaire spaceℕℕ, the infinite symmetric groupS∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc.In this theory sets are classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have theBorelsets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the correspondingBorel hierarchy(sets). After this come theprojective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the correspondingprojective hierarchy(sets).There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., theBorel(),analytic() andcoanalytic() sets.

Computable topological abelian groups

We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [3], [4], [5], [6], involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main results for the reals.

The class of C C -sets in a Polish space is the smallest σ \sigma -field containing the Borel sets and closed under operation ( A ) (\mathcal {A}) . In this article we show that any C C -set in the product of two Polish spaces can be approximated (in measure and category), uniformly over all sections, by sets generated by rectangles with one side a C C -set and the other a Borel set. Such a formulation unifies many results in the literature. In particular, our methods yield a simpler proof of a selection theorem for C C -sets with G δ {G_\delta } -sections due to Burgess [4].

Our purpose is to answer the question of Leader [1, p. 2341, by producing an example of a sigma algebra a of subsets of a set X and a universally a-integrable function f (i.e., ffdu exists for every countably additive set function u on a) which is not measurable with respect to a. We shall show that the sigma algebra a of Borel subsets of the interval [0, 1] admits nonmeasurable universally a-integrable functions (R. E. Zink suggested that this Borel algebra might yield an example). Lusin has shown [2] that, subject to the continuum hypothesis, there exists an uncountable subset V of X = [0, 1] such that each perfect nowhere dense subset G of X contains at most an enumerable set of points of V. Let H be a subset of V that is not a Borel set. We shall now show that the characteristic function x(H) of H is universally sigma integrable. To this end suppose u is a countably additive set function on a-. Then [5] u = c +j where c is continuous and j= Ijk where jk is a two valued jump function, the range of k is at most enumerable, and the variation of j is the sum of the variations of the jk'S; moreover, since the variation of u is the variation of j plus the variation c of c, each of j, c, and c is countably additive on a. Corresponding to each jk there is [4, Theorem 27.1] a point Xk such that jk(E) =jk(En [Xk ]) for every Borel subset E of X. Hence fX(H)dj= Ejk(HCn[Xk]) and it suffices to show that there exists a Borel subset B of X such that HCB and c(B) = 0 (then fX(H)dc= 0). There exists [3 ] a set K = U i Fi, F1 closed for i > 1, of the first category in X and a set L such that c(L) = 0 and KUL = X. For each positive integer i there exists a perfect set Gi and an at most enumerable set Hi such that Fi = Gi U Hi. Thus H n K = HC(Ui Fi) = Hf (Ui (GiUkHi)) = Ui (Hrl (GiU-Hi)) CUi((HnGi) UHi) is at most enumerable since every HrlGi (Fi is a closed subset of X and K is of the first category in X; hence Fi is nowhere dense in X which implies that G, is a perfect nowhere dense subset of X) and every H, is at most enumerable. Since e is countably additive on o and c vanishes on one point sets, c vanishes on countable sets. Hence H=Hr)(KUL) C(HnK)UL and c(HnK)UL) <?(HnK) +Z(L) =0.

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and Open image in new window stands for the one-point compactification of X, while EX is the equivalence relation which is defined on the Polish group C(X,R+*) by Open image in new window where f, g are in C(X,R+*), then EX is induced by a turbulent Polish group action. Second we show that given any Open image in new window if we identify the n-dimensional unit sphere Sn with the one-point compactification of Rn via the stereographic projection, while En,r is the equivalence relation which is defined on the Polish group Cr(Rn,R+*) by Open image in new window where f, g are in Cr(Rn,R+*), then En,r is also induced by a turbulent Polish group action.

This note deals with ( M , ∗ ) ({\mathbf {M}},\ast ) functions for various families M {\mathbf {M}} . It is shown that if M {\mathbf {M}} is the family of Borel sets of additive class α \alpha on a metric space X X , then ( M , ∗ ) ({\mathbf {M}},\ast ) functions are just the functions of the form sup y g ( x , y ) {\sup _y}g(x,y) where g : X × R → R g:X \times R \to R is continuous in y y and of class α \alpha in x x . If M {\mathbf {M}} is the class of analytic sets in a Polish space X X , then the ( M , ∗ ) ({\mathbf {M}},\ast ) functions dominating a Borel function are just the functions sup y g ( x , y ) {\sup _y}g(x,y) where g g is a real valued Borel function on X 2 {X^2} . It is also shown that there is an A A -function f f defined on an uncountable Polish space X X and an analytic subset C C of the real line such that f − 1 ( C ) ∉ {f^{ - 1}}(C) \notin the σ \sigma -algebra generated by the analytic sets on X X .

We consider a descriptive set-theoretic analog of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two aperiodic Borel automorphisms of uncountable Polish spaces. Using this, we classify all Borel ℝ -flows up to C^∞ -time-change isomorphism. We then extend the notion of descriptive Kakutani equivalence to all (not necessarily injective) Borel functions, and provide a variety of results leading towards a complete classification. The main technical tools are a series of Glimm–Effros and Dougherty–Jackson–Kechris-style embedding theorems.

It is shown that a measurable function from an atomless Loeb probability space (Ω, A, P) to a Polish space is at least continuum-to-one valued almost everywhere. It follows that there is no injective mapping h : [0, 1] → Ω such that h([a, b]) is Loeb measurable for each 0 ≤ a ≤ b ≤ 1 and P(h([0,1])) > 0. Thus, when an atomless Loeb measurable algebra on an internal set of cardinality continuum is imposed on the unit interval [0, 1] through a bijection, it cannot contain the Borel algebra.

We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(\Omega , \mu )$ admits an invariant $\sigma $-finite measure equivalent to $\mu $. Second, we prove the following de Finetti-type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^{M}$, where $Z$ is a Polish space, is a product measure.