Assume we are given compact subset $X \subset \mathbb{R}^n$. Consider the space of radon measures $M(X,\mathbb{R})$. I'm trying to find some Hilbert norm on this space either globally or locally. I use the term locally in the sense that for a bounded subset of $M$, the norm is Hilbertian.
I haven't been able to come across any Hilbertian norm for this space. One direction I tried to explore has been to study the weak* topological dual $C_0(X,R)$, i.e. the space of continuous functions on $X$ that vanish at infinity. Idea would be to say consider a kernel (symmetric) $k(\cdot,\cdot) : X \times X \to \mathbb{R}$ and use this to define a norm using the following inner-product like this:
$$\langle \mu, \nu\rangle := \int \int k(x,y) d\mu(x)\, d\nu(y)$$
Does this make sense? Please let me know if there are better well-known Hilbertian norms (local or global) on the space of measures.
