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Relations between Shy Sets and Sets of ν p -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

An example of a non-zero non-atomic translation-invariant Borel measure ν p on the Banach space ℓ p ( 1 ≤ p ≤ ∞ ) is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " ν p -almost every element of ℓ p has a property P" implies that “almost every” element of ℓ p (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

Byoung Soo Kim, Dong Hyun Cho (2017)

Czechoslovak Mathematical Journal

Let C [ 0 , t ] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [ 0 , t ] , and define a random vector Z n : C [ 0 , t ] → ℝ n + 1 by Z n ( x ) = x ( 0 ) + a ( 0 ) , ∫ 0 t 1 h ( s ) d x ( s ) + x ( 0 ) + a ( t 1 ) , ⋯ , ∫ 0 t n h ( s ) d x ( s ) + x ( 0 ) + a ( t n ) , where a ∈ C [ 0 , t ] , h ∈ L 2 [ 0 , t ] , and 0 < t 1 < ⋯ < t n ≤ t is a partition of [ 0 , t ] . Using simple formulas for generalized conditional Wiener integrals, given Z n we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra 𝒮 . Finally, we express the generalized analytic conditional Feynman...

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