1 Probability measures on metric spaces 1 1.1 Borel sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Borel probability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Narrow convergence of measures . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The bounded Lipschitz metric . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5...

The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. I...

The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. I...

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i = 1, . . . , N , in some Hausdorff space Y . There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X ⊂ Y . Typically, some kind of contractivity...

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i = 1, . . . , N , in some Hausdorff space Y . There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X ⊂ Y . Typically, some kind of contractivity...

We obtain some results about Borel maps with meager fibers on completely metrizable separable spaces. The results are related to a recent dichotomy by Sabok and Zapletal, concerning Borel maps and σ-ideals generated by closed sets. In particular, we give a “classical” proof of this dichotomy. We shall also show that for certain natural σ-ideals I generated by closed sets in compact metrizable s...

We first consider infinite two-player games on pushdown graphs. In previous work, Cachat, Duparc and Thomas [4] have presented a winning decidable condition that is Σ3-complete in the Borel hierarchy. This was the first example of a decidable winning condition of such Borel complexity. We extend this result by giving a family of decidable winning conditions of arbitrary finite Borel complexity....

(A) We consider measure preserving actions of an infinite, countable (discrete) group Γ on non-atomic standard measure spaces (X,μ), i.e., standard Borel spaces equipped with a non-atomic probability Borel measure. (All such measure spaces are isomorphic to ([0, 1], λ), where λ is Lebesgue measure.) We denote by A(Γ, X, μ) the space of such actions. If a ∈ A(Γ, X, μ) and γ ∈ Γ, we denote by γ(x...

We consider Borel sets of finite rank A C A'" where cardinality of A is less than some uncountable regular cardinal K5. We obtain a "normal form" of A, by finding a Borel set Q, such that A and Q continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum. to multiplication by a countable ordinal, and to ordinal exponentiati...

The paper is devoted to the study of topologies on the group Aut(X, B) of all Borel automorphisms of a standard Borel space (X, B). Several topologies are introduced and all possible relations between them are found. One of these topologies, τ , is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of Aut(X, B) and find their closures. ...