We introduce the concept of Mσ-approachability for a semicontinuous function (i.e. a nonadditive measure) on a σ-complete sublattice of a locally complete σ-continuous lattice L and using it we extend a nonadditive measure from a sublattice of L to a σ-complete sublattice of L. AMS Mathematics Subject Classification (2000): 28A12, 28C15, 28B10

We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these. Introduction Let (X,B, m, T ) be an invertible, ergodic measure preserving transformation of a σ-finite measure space, then there are no other σ-finite, m-absolu...

Let Σ be a ranked set. A categorical Σ-algebra, cΣa for short, is a small category C equipped with a functor σC : C n //C, for each σ ∈ Σn, n ≥ 0. A continuous categorical Σ-algebra is a cΣa which has an initial object and all colimits of ω-chains, i.e., functors N //C; each functor σC preserves colimits of ω-chains. (N is the linearly ordered set of the nonnegative integers considered as a cat...

The celebrated Kadison-Sakai Theorem states that all derivations on von Neumann algebras are inner. In this paper, we generalize this theorem to the so-called ∗-σderivations where σ is a weakly continuous ∗-linear mapping. We decompose a σ-derivation into a sum of an inner σ-derivation and a multiple of a homomorphism. As a consequence, we establish an extension of the Singer-Wermer Theorem. A ...

Let ∆ ⊆ C be the open unit disc and let Σ ⊆ ∆×∆ be a compact set such that K = Σ ∪ (∂∆×∆) is a connected set. It is a classical result by Hartogs that if Σ is an analytic variety over ∆ with the boundary in ∂∆×∆, then every function holomorphic in a connected neighbourhood of K extends holomorphically to a neighbourhood of ∆ × ∆. It is proved that the same conclusion holds if Σ is a ‘continuous...

Let σ be a probability Borel measure on the unit circle T and {φn} be the orthonormal polynomials with respect to σ. We say that σ is a Szegő measure, if it has an arbitrary singular part σs, and R T log σ ′dm > −∞, where σ′ is the density of the absolutely continuous part of σ, m being the normalized Lebesgue measure on T. The entropy integrals for φn are defined as n = Z T |φn| log |φn|dσ It ...

lorentz characterized the almost convergence through the concept of uniform convergence of de lavallée-poussin mean. in this paper, we generalize the notion of almost convergence by using the concept ofinvariant mean and the generalized de la vallée-poussin mean. we determine the bounded linear operators forthe generalized σ-conservative, σ-regular and σ-coercive matrices.

We present a categorical approach to the extension of probabilities, i.e. normed σ-additive measures. J. Novák showed that each bounded σ-additive measure on a ring of sets is sequentially continuous and pointed out the topological aspects of the extension of such measures on over the generated σ-ring σ( ): it is of a similar nature as the extension of bounded continuous functions on a complete...

In the last several lectures, we have discussed controller synthesis for: • Discrete Systems: design is characterized by a fixed point of a difference equation • Continuous Systems: design is characterized by a fixed point of a partial differential equation In each case, the solution is characterized as a fixed point of an equation, which we refer to as the Hamilton-Jacobi equation. In these le...

In the last two weeks, we have discussed controller synthesis for: • Discrete Systems: design is characterized by a fixed point of a difference equation • Continuous Systems: design is characterized by a fixed point of a partial differential equation In each case, the solution is characterized as a fixed point of an equation, which we refer to as the Hamilton-Jacobi equation. In these lecture n...