Regularization of closed-valued multifunctions in a non-metric setting
An example of a non-zero non-atomic translation-invariant Borel measure on the Banach space is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition "-almost every element of has a property P" implies that “almost every” element of (in the sense of [4]) has the property P. It is also shown that the converse is not valid.
Let denote a generalized Wiener space, the space of real-valued continuous functions on the interval , and define a random vector by where , , and is a partition of . Using simple formulas for generalized conditional Wiener integrals, given we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra . Finally, we express the generalized analytic conditional Feynman...
We show that for every ergodic flow, given any factor σ-algebra ℱ, there exists a σ-algebra which is relatively perfect with respect to ℱ. Using this result and Ornstein's isomorphism theorem for flows, we give a functorial definition of the entropy of flows.
Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has . Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.