The Riesz's representation theorem says that every bounded linear functional $L$ on $C \,[a,b]$ can be representated by a Riemann - Stieltjes integral: $$L(f) = \int_{a}^{b}f(x)d(\alpha(t))$$ where $\alpha$ is a bounded variation on $[a,b]$.
So, i have the following linear functional $L: C\,[a,b]\rightarrow\mathbb{R};\;L(f) =f(x_0)$ with $x_0 \in [a,b]$ fixed.
How i can find the bounded variation function $\alpha$ ?
Define $\alpha: [a,b] \to \Bbb{R}$ by $$ \alpha(t):= \begin{cases} 0 & \text{ if } a \leq t <x_0 \\ 1 & \text{ if } x_0 \leq t \leq b \end{cases} $$ Then, $$ \int_{a}^b f d\alpha = f(x_0) $$