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.

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

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 $Z_n:C[0,t]\to {ℝ}^{n+1}$ by $$Z_n\left(x\right)=\left(x\left(0\right)+a\left(0\right),{\int }_0^{t_1}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_1\right),\cdots ,{\int }_0^{t_n}h\left(s\right)\mathrm{d}x\left(s\right)+x\left(0\right)+a\left(t_n\right)\right),$$ where $a\in C[0,t]$, $h\in L_2[0,t]$, and $0<t_1<\cdots <t_n\le t$ 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...

On établit pour le cône $C$ des mesures $\mu$ positives bornées sur $𝒞\left(\mathbf{R}\right)$, quasi-invariantes sous les translations de $𝒟\left(\mathbf{R}\right)$ et vérifiant :$$\mu (f+dw)=\mu \left(dw\right)\exp {\int }_{R}dt\left[\right(w\left(t\right)+\frac{1}{2}f\left(t\right))f^{\text{'}\text{'}}\left(t\right)-P\left(w\right(t)+f\left(t\right)+P\left(w\right(t\left)\right)]$$(avec $P$ polynôme borné inférieurement) les résultats suivants :– Toute mesure de $C$ est intégrale de mesures appartenant aux génératrices extrémales de $C$.– Les génératrices extrémales de $C$ sont composées de mesures markoviennes.