On constructing finite, finitely subadditive outer measures, and submodularity.
On continuity at zero of the maximal operator for a semifinite measure
For a sequence of linear maps defined on a Banach space with values in the space of measurable functions on a semifinite measure space, we examine the behavior of its maximal operator at zero.
On continuity of measurable group representations and homomorphisms
Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space ℒ(H) of bounded linear operators on H with the weak operator topology. We prove that if U is a measurable map from G to ℒ(H) then it is continuous. This result was known before for separable H. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.
On continuous collections of measures
An integral representation theorem is proved. Each continuous function from a totally disconnected compact space to the probability measures on a complete metric space is shown to be the resolvent of a probability measure on the space of continuous functions from to .
On continuous interval functions
On control submeasures and measures
On convergences of signed states
On c-sets and products of ideals
Let X, Y be uncountable Polish spaces and let μ be a complete σ-finite Borel measure on X. Denote by K and L the families of all meager subsets of X and of all subsets of Y with μ measure zero, respectively. It is shown that the product of the ideals K and L restricted to C-sets of Selivanovskiĭ is σ-saturated, which extends Gavalec's results.
On differential bases formed of intervals.
On differentiation of integrals with respect to bases of convex sets.
Differentiation of integrals of functions from the class with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in , N ≥ 3, and with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension.
On dimensions of semimetrized measure spaces
On distribution functions of the derivatives of weakly differentiable mappings
On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process
We study the Hausdorff dimension of measures whose weight distribution satisfies a Markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the packing dimensions equal the upper Rényi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when viewed as a function of the distributed weights under the norm.
On essential cluster sets
On extension of Baire submeasures
On extension of submeasures
On extensions of measures
On extensions of -fields of sets
On fields and ideals connected with notions of forcing
We investigate an algebraic notion of decidability which allows a uniform investigation of a large class of notions of forcing. Among other things, we show how to build σ-fields of sets connected with Laver and Miller notions of forcing and we show that these σ-fields are closed under the Suslin operation.
On finitely subadditive outer measures and modularity properties.