This note is about functions f : Aω → Bω whose graph is recognized by a Büchi finite automaton on the product alphabet A×B. These functions are Baire class 2 in the Baire hierarchy of Borel functions and it is decidable whether such fonction are continuous or not. In 1920 W. Sierpinski showed that a function f : R → R is Baire class 1 if and only if both the overgraph and the undergraph of f ar...

We study the minimization problem f (x) → min, x ∈ C, where f belongs to a complete metric space of convex functions and the set C is a countable intersection of a decreasing sequence of closed convex sets Ci in a reflexive Banach space. Let be the set of all f ∈ for which the solutions of the minimization problem over the set Ci converge strongly as i→∞ to the solution over the set C. In our r...

If X is a metric space then CX and LX denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of CX modulo LX is the least cardinality of any set U \LX where U generates CX . For a large class of separable metric spaces X we prove that the relative rank of CX modulo LX is uncountable. When X is the Baire space NN, this rank is א1. A large pa...

Let (Ω,B) be a measurable space, An ⊂ B a sub-σ-field and μn a random probability measure on (Ω,B), n ≥ 1. In various frameworks, one looks for a probability P on B such that μn is a regular conditional distribution for P given An for all n. Conditions for such a P to exist are given. The conditions are quite simple when (Ω,B) is a compact Hausdorff space equipped with the Borel or the Baire σ-...

We generalize the Baire Category Theorem to the Borel and difference hierarchies, i.e. if Γ is any of the classes Σξ , Π 0 ξ , Dη(Σ 0 ξ) or Ďη(Σ 0 ξ) we find a representative set PΓ ∈ Γ and a Polish topology τΓ such that for every A ∈ Γ̌ from some assumption on the size of A ∩ PΓ we can deduce that A \ PΓ is of second category in the topology τΓ. This allows us to distinguish the levels of the B...

We show that every array (x(i; j) : 1 i < j < 1) of elements in a point-wise compact subset of the Baire-1 functions on a Polish space, whose iterated pointwise limit lim i lim j x(i; j) exists, is converging Ramsey-uniformly. An array (x(i; j) i<j) in a Hausdorr space T is said to converge Ramsey-uniformly to some x in T , if every subsequence of the positive integers has a further subsequence...

In this paper, we study different types of non-additive set multifunctions (such as: uniformly autocontinuous, null-additive, null-null-additive), presenting relationships among them and some of their properties regarding atoms and pseudo-atoms. We also study non-atomicity and non-pseudo-atomicity of regular null-additive set multifunctions defined on the Baire (Borel respectively) δ-ring of a ...

For a free ultrafilter U on ω we study several cardinal characteristics which describe part of the combinatorial structure of U . We provide various consistency results; e.g. we show how to force simultaneously many characters and many π–characters. We also investigate two ideals on the Baire space ωω naturally related to U and calculate cardinal coefficients of these ideals in terms of cardina...

We present a general way of defining various reduction games on ω which “represent” corresponding topologically defined classes of functions. In particular, we will show how to construct games for piecewise defined functions, for functions which are pointwise limit of certain sequences of functions and for Γ-measurable functions. These games turn out to be useful as a combinatorial tool for the...

We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions. The starting point for writing this paper was the desire to improve the results of V.K. Maslyuchenko et al. [MMMS], [MS], [KM], [KMM] who generalized a classical theorem of W.Rudin [Ru]...