Let $f$ be a function defined on a $J=[0,1]$, which is $k$-times differentiable on $J$; i.e. $f\in C^k(J)$.
By the Riesz representation theorem we know that for any complex Radon measure $\mu$ on $J$ , we have the isometric vector space isomorphism between the space of complex Radon measures on $J$ and the space of bounded linear functionals, given by $\mu\mapsto \int f \ d\mu.$
Given a bounded linear functional $I$ (which we know is of the form $\int f\ d\nu$) on $C^k(J)$, I want to show that we can find a unique $\mu, a_0,\dots,a_{k-1}$ where $\mu$ is a complex Borel measure and the $a_i$ are constants, with
$I(f) = \int_{J} f\ d\nu = \int_J f^{(k)}\ d\mu + \sum_{i=0}^{k-1} a_i f^{(i)}(0)$
This is also found as an exercise in Folland, 7.27, if that is helpful. Any guidance would be greatly appreciated.