concerned with the Banach spaces $(\S^{w},\Vert\cdot\Vert_{2})$ and $(\S^{s},\Vert\cdot\Vert_{\infty})$. Now fix $* \in \{s,w\}$; a function $f$ belonging to the set $C([0,1] )$ of continuous function on $[0,1] $ is said to lie in $\mathcal{D}^*(\Delta_{\nu,\mu})$ if there exist $a,b \in \R$ and $g \in \S^*$ with $$\begin{aligned}
\label{KreinFeller}
 f(x)=a+bF_{\mu}(x)+\int_{[0,x]}( F_{\mu}(x)-F_{\mu}(y) ) g(y)\;{\mathrm{d}}\nu(y)
 \end{aligned}$$ for all $x \in [0,1]$. Alternatively, by Fubini we can write $ f(x)=a+bF_{\mu}(x)+\int_{[0,x]}\int_{[0,y]}  g(s)\;{\mathrm{d}}\nu(s)\;{\mathrm{d}}\mu(y)$. By the uniqueness of densities we observe that $f$ determines $a$, $b$ and $g$ uniquely and by setting $ \Delta_{\nu,\mu}f \coloneqq g$ we define an injective linear operator $\Delta_{\nu,\mu}:\mathcal{D}^*(\Delta_{\nu,\mu})\to \S^*$. The first derivative of $ f \in \mathcal{D}^{\text{*}}(\Delta_{\nu,\mu})$ is defined by $$\begin{aligned}
\label{eq:first_derivative}
\nabla_{\mu}f(x) \coloneqq \nabla_{\mu}f(0)+\int_{[0,x]} \Delta_{\nu,\mu}f(y)\;{\mathrm{d}}\nu(y),
\;\; \text{where} \;\;
\nabla_{\mu}f(0) \coloneqq \lim_{x \downarrow 0^+} \dfrac{f(x)-f(0)}{F_{\mu}(x)-F_{\mu}(0)}
\;\;
\text{and}
\;\;
 0^+ \coloneqq \inf ({\operatorname{supp}}(\mu)).\end{aligned}$$ The existence of the above limit follows from and the assumption that ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}( \mu)$. Additionally, from together with Lebesgue’s dominated convergence theorem, we have that $f$ is constant on every interval of constancy of $F_{\mu}$, $a=f(0)$, $b= \nabla f_{\mu}(0)$ and $\mathcal{D}^s ( \Delta_{\nu,\mu} ) \subset \mathcal{D}^w (\Delta_{\nu,\mu}) \subset C_{\nu,\mu}$. Moreover, for all $f \in\mathcal{D}^*(\Delta_{\nu,\mu})$ and all $x \in (0,1)$, $$\begin{aligned}
    \nabla_{\mu}(f)(x)
    = \lim_{ \substack{ y \downarrow x^+}} \dfrac{f(y)-f(x)}{F_{\mu}(y)-F_{\mu}(x)}   +  