On the descriptive sot theory of the lexicographic square
On the descriptive theory of sets
On the difference property of Borel measurable functions
If an atomlessly measurable cardinal exists, then the class of Lebesgue measurable functions, the class of Borel functions, and the Baire classes of all orders have the difference property. This gives a consistent positive answer to Laczkovich's Problem 2 [Acta Math. Acad. Sci. Hungar. 35 (1980)]. We also give a complete positive answer to Laczkovich's Problem 3 concerning Borel functions with Baire-α differences.
On the difference property of families of measurable functions
We show that, generally, families of measurable functions do not have the difference property under some assumption. We also show that there are natural classes of functions which do not have the difference property in ZFC. This extends the result of Erdős concerning the family of Lebesgue measurable functions.
On the differentiation of integrals of functions from Lφ(L)
On the differentiation of integrals of functions from Orlicz classes
On the differentiation theorem in metric groups
On the dimension of an irrigable measure
On the dimension of limit sets of geometrically finite Möbius groups.
On the dimension of -harmonic measure in space
Let , and let , be given. In this paper we study the dimension of -harmonic measures that arise from non-negative solutions to the -Laplace equation, vanishing on a portion of , in the setting of -Reifenberg flat domains. We prove, for , that there exists small such that if is a -Reifenberg flat domain with , then -harmonic measure is concentrated on a set of -finite -measure. We prove, for , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of -harmonic measure...
On the dimensions of certain incommensurably constructed sets.
On the double -integral.
On the duality between -modulus and probability measures
Motivated by recent developments on calculus in metric measure spaces , we prove a general duality principle between Fuglede’s notion [15] of -modulus for families of finite Borel measures in and probability measures with barycenter in , with dual exponent of . We apply this general duality principle to study null sets for families of parametric and non-parametric curves in . In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence...
On the dynamical meaning of spectral dimensions
On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation
On the equivalence between CH and the existence of certain -Luzin subsets of .
On the Euler Characteristic in Spaces with a Separability Property.
On the existence of conditional density functions
On the existence of disintegrations
On the existence of measures on σ-algebras