I am a bit new to functional analysis and I stumbled upon this problem that confuses me.
Consider the space $X$ of bounded, non-decreasing, right-continuous functionals $F$ on $[0,1]$. Specifically, the space of cumulative distribution functions on $[0,1]$ is a subset of $X$. I have a linear functional $\varphi$ defined on $X$ through a set of properties. Intuitively (based on my knowledge of the specific function $\varphi$), it seems obvious that it should be of the form $$ \varphi(F) = \int_0^1 G_\varphi(z) dF(z) $$ for some function $G_\varphi$. Also, it seems obvious that $G_\varphi \in X$.
Now, I am trying to prove this using the Riesz representation theorem. For this, we need an inner product on $X$. My first attempt was to define an "inner product" $\langle \cdot, \cdot \rangle : X \times X \to \mathbb{R}$ of the form $$ \langle F, G \rangle = \int_0^1 G(z) dF(z)$$ If I blindly apply the Riesz representation theorem using this "inner product" and recover "Riesz representer" $G_\varphi$ using properties of $\varphi$, then I get exactly the answer I expect. The problem, however, is that $\langle \cdot, \cdot \rangle$ defined above is not an inner product, as it is not symmetric.
An alternative would be to move to a space $Y$ in which the derivatives of $F \in X$ live (i.e., define $\varphi$ as a function of the pdf $f \in Y$ corresponding to the original cdf $F \in X$ instead), which has the suitable inner product $$ \langle f, g \rangle = \int_0^1 f(z) g(z) dz.$$ The problem is that the cdf $F(z)$ in my problem does not necessarily have a derivative $f (z)$, especially at $z = 1$.
Does anyone know whether there exists a version of the Riesz representation theorem that can deal with the stieltjes integral above? Or are there other creative suggestions? Thanks in advance!