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The main concern of this paper is to present some improvements
to results on the existence or non-existence of countably additive Borel measures
that are not Radon measures on Banach spaces taken with their weak topologies, on
the standard axioms (ZFC) of set-theory. However, to put the results in perspective we
shall need to say something about consistency results concerning measurable cardinals.
For a Banach space and a probability space , a new proof is given that a measure , with , has RN derivative with respect to iff there is a compact or a weakly compact such that is a finite valued countably additive measure. Here we define where is a finite disjoint collection of elements from , each contained in , and satisfies . Then the result is extended to the case when is a Frechet space.
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...
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