In my work, I used recently the classical Riesz theorem. It has lead me to study functions of bounded variations and Riemann-Stieltjes integrals. Unfortunately, even if there exist a lot of books and references about these subjects, I do not find clearly the answers to my multiple questions.
Actually, I think that all my questions have positive answers, but it is just instinct and I would like to have now rigorous proofs of the following elementary properties. I warmly thank everyone for considering my request.
Let $\varphi$ be a continuous linear form on $C([0,T],\mathbb{R})$. From the classical Riesz theorem, there exists a unique (up to an additive constant) function of bounded variations $\alpha : [0,T] \to \mathbb{R}$ such that $$ < \varphi , f > = \int_0^T f(x) d\alpha (x) $$ for every $f \in C([0,T],\mathbb{R})$, where the above integral is the classical Riemann-Stieltjes integral.
My questions are the following (the most important being the fourth one):
1) I know that $< \varphi , f > \geq 0$ for every nonnegative functions $f \in C([0,T],\mathbb{R}^+_0)$. Can we conclude that $\alpha$ is a nondecreasing function?
2) Let $f \in C([0,T],\mathbb{R})$ and let $p : [0,T] \to \mathbb{R}$ be defined by $$ p(t) = \int_0^t f(x) d\alpha (x) $$ for every $t \in [0,T]$. Can we conclude that $p$ is a function of bounded variations?
3) If yes, can we conclude that $dp = fd\alpha$ in the sense that $$ \int_0^T g(x) dp(x) = \int_0^T g(x) f(x) d\alpha (x) $$ for every $g \in C([0,T],\mathbb{R})$?
4) Finally, I wonder if the following Fubini formula holds: $$ \int_0^T \int_0^t g(t,s) ds d\alpha (t) = \int_0^T \int_s^T g(t,s) d\alpha(t) ds $$ where $g : [0,T]^2 \to \mathbb{R}$ is bounded and continuous in the first variable.
If possible, I would like to have answers in terms of Riemann-Stieltjes integral, and not Lebesgue-Stieltjes integral. I warmly thank you for all answers or references...