An exact Radon-Nikodym derivative is obtained for a pair (I,J) of positive linear functionals, with J absolutely continuous with respect to I, using a notion of exhaustion of I on elements of a function algebra lattice.

Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_{\varphi }$ of Wiener measure and for a partition $0=t_0<t_1<\cdots <t_n<t_{n+1}=T$ of $[0,T]$, let $X_{n}C[0,T]\to {ℝ}^{n+1}$ and $X_{n+1}C[0,T]\to {ℝ}^{n+2}$ be given by $X_n\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_n\right))$ and $X_{n+1}\left(x\right)=(x\left(t_0\right),x\left(t_1\right),\cdots ,x\left(t_{n+1}\right))$, respectively. In this paper, using a simple formula for the conditional $w_{\varphi }$-integral of functions on $C[0,T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_{\varphi }$-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_{\varphi }$-integral...

Let $T$ be a locally compact Hausdorff space and let $C_0\left(T\right)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of $X$-valued $\sigma$-additive Baire measures on $T$ is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map $u{C}_0\left(T\right)\to X$ when...