Some examples of d-ideals and related Baire systems
On the Baire system generated by a linear lattice of functions
σ-ideals and related Baire systems
On rectangles and countably generated families
A representation theorem for the second dual of C[0,1]
Some complexity results in topology and analysis
If X is a compact metric space of dimension n, then K(X), the n- dimensional kernel of X, is the union of all n-dimensional Cantor manifolds in X. Aleksandrov raised the problem of what the descriptive complexity of K(X) could be. A straightforward analysis shows that if X is an n-dimensional complete separable metric space, then K(X) is a or PCA set. We show (a) there is an n-dimensional continuum X in for which K(X) is a complete set. In particular, ; K(X) is coanalytic but is not an analytic...
Some additive properties of sets of real numbers
On partitions of lines and space
We consider a set, L, of lines in and a partition of L into some number of sets: . We seek a corresponding partition such that each line l in meets the set in a set whose cardinality has some fixed bound, . We determine equivalences between the bounds on the size of the continuum, , and some relationships between p, and .
On infinite partitions of lines and space
Given a partition P:L → ω of the lines in , n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, , so that each line meets at most m points of its color. Assuming Martin’s Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations...
Projective well-orderings and extensions of Lebesgue measure
Zobecnění Velké Fermatovy věty: Bealova domněnka a problém o cenu
Conformal measures for rational functions revisited
We show that the set of conical points of a rational function of the Riemann sphere supports at most one conformal measure. We then study the problem of existence of such measures and their ergodic properties by constructing Markov partitions on increasing subsets of sets of conical points and by applying ideas of the thermodynamic formalism.
Counting and convergence in ergodic theory
Borel equivalence and isomorphism of coanalytic sets
CONTENTSIntroduction...............................................................51. Coanalytic sets and admissible ordinals...............72. The hypothesis of constructibility........................123. Ordinal partitions and non-isomorphic sets.........164. Thin non-isomorphic sets....................................195. The hypothesis of projective determinacy...........226. Further results and open questions....................25References.............................................................28...
On the Borel class of the derived set operator
Problem 24 of the "Scottish book'' concerning additive functionals
On the Borel class of the derived set operator. II
Which Bernoulli measures are good measures?
For measures on a Cantor space, the demand that the measure be "good" is a useful homogeneity condition. We examine the question of when a Bernoulli measure on the sequence space for an alphabet of size n is good. Complete answers are given for the n = 2 cases and the rational cases. Partial results are obtained for the general cases.
Some comments on infinite Boolean functions
We introduce infinite Boolean functions and investigate some of their properties.
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