On Compactness and Convergence in Spaces of Measures.
On compactness and extreme points of some sets of quasi-measures and measures. II.
On Compactness and Extreme Points of Some Sets of Quasi-Measures and Measures.
On compactness of lattices.
On complete measurability of multifunctions defined on product spaces.
On complete-cocomplete subspaces of an inner product space
In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space is complete if and only if there exists a -additive state on , the orthomodular poset of complete-cocomplete subspaces of . We then consider the problem of whether every state on , the class of splitting subspaces of , can be extended to a Hilbertian state on ; we show that for the dense hyperplane (of a separable Hilbert space) constructed by P. Pták and...
On completion of measures on a --ring
On complex Radon measures. I
On concentration of self-bounding functions.
On Conditions for Unrectifiability of a Metric Space
We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.
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.