A Borel topos
A Cantor regular set which does not have Markov's property
A category analogue of the density topology
A category Ψ-density topology
Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open...
A class of Caratheodory outer measures in topological spaces
A class of sets whose distance set fills an interval
A Convergence Theorem for Measures in Regular Hausdorff Spaces.
A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections
Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then is a set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of...
A covering property of some classes of sets in
A dominated convergence theorem for real-valued summable set functions
A general form of the Vitali theorem
A generalization of the Nikodym boundedness theorem.
In this note an internal property of a ring of sets, named the Nested Partition Property, is shown to imply the Nikodym Property. A wide range of examples are shown to have this property.
A generalization of the Vitali covering theorem
A measurable selection theorem
A metric space associated with a probability space.
A new type of affine Borel function.
A Note on a Theorem of A. D. Alexandroff
A note on complex unions of subsets of the real line
A note on determinacy of measures
A note on ideals of compact sets
Solecki has shown that a broad natural class of ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed subset in this representation can be taken to be closed upwards.