I am trying to understand a proof of a variant of the Riesz representation theorem.
Consider a linear functional $l$ on the space of continuous functions on $[0,1]$. Assume further that $\ell(f)\ge 0$ when $f\ge0$ on $[0,1]$.
We are going to define a function $F(u)$ on $[0,1]$ that will be used to construct a Lebesgue-Stieltjes measure. Define a collection of auxiliary functions $f_\epsilon(x,u)$ so they are $1$ on $[0,u]$, $0$ on $[u+\epsilon, 1]$, and linearly interpolated between those two intervals. The graph from left to right of such a function is a horizontal line with $y=1$, a downward sloping line, and then a horizontal line along 0.
Define $F(u)=\lim_{\epsilon\rightarrow 0}\ \ell(f_\epsilon(x,u)).$
We see that $F$ is increasing, and as shown in the answers, right-continuous. Why do we have
$\ell(f)=\int_0^1 f(x)\ dF(x)$
where the above is a Lebesgue integral.