The Baire and Borel measure
The Banach–Mazur game and σ-porosity
It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.
The coarsest topology for -approximately continuous functions
The complexity of the set of squares in the homeomorphism group of the circle
The set of squares in the group of autohomeomorphisms of the circle is complete analytic, and hence analytic but not Borel.
The Convex Hull of a Typical Compact Set.
The covering property for σ-ideals of compact, sets
The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of .
The Existence of Measurable Approximating Maximums.
The Hahn-Banach theorem implies the Banach-Tarski paradox
The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set
The Hausdorff dimension of some special plane sets
A compact set is constructed such that each horizontal or vertical line intersects in at most one point while the -dimensional measure of is infinite for every .
The kernel operation on subsets of a -space
The max norm in -isometries and measure
The measure extension problem for vector lattices
Let be a boundedly -complete vector lattice. If each -valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a -additive measure on the generated -field then is said to have the measure extension property. Various sufficient conditions on which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: has the...
The pathological infinity of measures
The prevalence of permutations with infinite cycles
A number of recent papers have been devoted to the study of prevalence, a generalization of the property of being of full Haar measure to topological groups which need not have a Haar measure, and the dual concept of shyness. These concepts give a notion of "largeness" which often differs from the category analogue, comeagerness, and may be closer to the intuitive notion of "almost everywhere." In this paper, we consider the group of permutations of natural numbers. Here, in the sense of category,...
The product of certain measurable spaces
The Ramsey sets and related sigma algebras and ideals
The Range of Atomless Group Valued Measures
The structure of the -ideal of -porous sets
We show a general method of construction of non--porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non--porous Suslin subset of a topologically complete metric space contains a non--porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non--porous element. Namely, if we denote the space of all compact subsets of a compact metric space with the Vietoris topology...
The σ-ideal of closed smooth sets does not have the covering property
We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.