TITLE: Borel Measure such that integrating a polynomial yields the derivative at a point
QUESTION [7 upvotes]: Does there exist a signed regular Borel measure such that
$$ \int_0^1 p(x) d\mu(x) = p'(0) $$
for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure at a point. If it were instead asking for the integral to yield $p(0)$, I would suggest letting $\mu = \delta_0$. That is, $\mu(E) = 1$ iff $0 \in E$. However, this is slightly different and I'm a bit unsure of it. It's been a while since  I've done any real analysis, so I've forgotten quite a bit. I took a look back at my old textbook and didn't see anything too similar. If anyone could give me a pointer in the right direction, that would be great. I'm also kind of curious if changing the integration interval from [0,1] to all of $\mathbb{R}$ changes anything or if the validity of the statement is altered by allowing it to be for all polynomials, instead of just polynomials of at most some degree.
Thanks!

REPLY [2 votes]: If you want a quick, lazy answer without any computation:
Consider the Banach space $C([0,1])$ and its $n+1$-dimensional subspace $\mathcal{P}_n$ consisting of all polynomials of degree at most $n$.  The map $\ell : \mathcal{P}_n \to \mathbb{R}$ defined by $\ell(p) = p'(0)$ is linear, so since $\mathcal{P}_n$ is finite dimensional, $\ell$ is continuous.  By the Hahn-Banach theorem it has a continuous linear extension $\tilde{\ell} : C([0,1]) \to \mathbb{R}$.  By the Riesz representation theorem there is a signed regular Borel measure $\mu$ on $[0,1]$ such that $\tilde{\ell}(f) = \int f\,d\mu$, and for polynomials $p$ we have $\int p\,d\mu = \tilde{\ell}(p) = \ell(p) = p'(0)$.