A theorem on the existence of separable supports of σ-finite Borel measures given on metric spaces with small topological weights is applied to constructions of certain analogues of special subsets of the real line. A well-known result from topological measure theory states that every σ-finite Borel measure on a metric space whose weight is not real-valued measurable possesses a separable suppo...

New Vapnik–Chervonenkis type concentration inequalities are derived for the empirical distribution of an independent random sample. Focus is on maximal deviation over classes Borel sets within a low probability region. The constants explicit, enabling numerical comparisons.

6. Suppose X is a metric space, let BX be the σ-algebra generated by the open sets of X, we call BX the Borel σ-algebra and its elements Borel sets. 7. A countable intersection of open sets is called a Gδ set ; a countable union of closed sets is called an Fσ set . 8. We call a collection of subsets of X an elementary family if (i) ∅ ∈ E ; (ii) if E,F ∈ E , then E ∩ F ∈ E ; and (iii) if E ∈ E ,...

We give in this paper additional answers to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:”A Decade of Concurrency”, Springer LNCS 803 (1994), 583-621], proving topological properties of omega context free languages (ω-CFL) which extend those of [O. Finkel, Topological Properties of Omega Context Free Languages, Theoretical Computer Science, Vol. 262 (1-2),...

For $S_g(x,y)=x-g(y), x,y\in\mathbb{R}^n, g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given $A$, both for general Borel subsets $\mathbb{R}^{2n}$ product sets.

Let B be a Borel set in E with volume V (B) = ∞. It is shown that almost all lattices L in E contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in E with V (S) = ∞. Then for almost all lattices L in E the successive minima λ1(S,L), . . . , λd(S,L) of S with respect to L are 0...

The main goal of this paper is to generalize several results concerning cardinal invariants to the statements about the associated families of sets. We also discuss the relationship between the additive properties of sets and their Borel images. Finally, we present estimates for the size of the smallest set which is not strongly meager.