Let $M$ be the positive borel measures on a hausdorff topological space $X$, which are finite on compacts sets $--$ i.e. the real cone of radon measures.
I am given a definition of a derivative of two radon measures, say $\mu, \eta \in M$, as follows.
Let
$D^+_\mu \eta (x) := \left\{ \begin{array}{1 1} \limsup \limits_{r \rightarrow 0} \frac{ \eta(\overline{B_r(0)}) }{ \mu(\overline{B_r(0)}) } & \mu(\overline{B_r(0)}) > 0 \forall r > 0\\ \infty & \mu(\overline{B_r(0)}) = 0 \forall r > 0 \end{array} \right. $
and
$D^-_\mu \eta (x) := \left\{ \begin{array}{1 1} \liminf \limits_{r \rightarrow 0} \frac{ \eta(\overline{B_r(0)}) }{ \mu(\overline{B_r(0)}) } & \mu(\overline{B_r(0)}) > 0 \forall r > 0\\ \infty & \mu(\overline{B_r(0)}) = 0 \forall r > 0 \end{array} \right. $
and we write in case of equality
$D_\mu \eta (x) := D^+_\mu \eta (x) = D^-_\mu \eta (x)$
I could accept this ad-hoc definition "as is". Nevertheless, I know there is a canonical topology on the linear space of bounded radon measures, so is probably on the cone of positive (not necessarily bounded) radon measures (Furthermore, I guess the thoughs pass over the linear space of signed Radon measures)
Hence, I would like to obtain a good intuition how the definition of derivative of above relates to the canonical definition (if applicable) of derivative on infinite-dimensional vector spaces. (I suppose there is such a connection). Can give a explanation or tell me a good tight resource to look this up?
Thank you.
I think there is some misunderstanding here. While it is possible to have derivatives of functions defined between Banach spaces (say) (e.g. with the concept of Frechet- or Gateaux-derivative), the derivative of a Radon measure is a derivative of an object in such a space. Hence, the derivative of a Radon measure is somewhat more related to the distributional derivative of a function or the weak derivative.
To get an impression of the derivative you are talking about, the following example may be helpful: Let $\mu$ be the one dimensional Lebesgue measure on $[0,1]$. For a integrable and positive function $f:[0,1] \to \mathbf{R}$ and define a measure $\nu$ by $\nu(A) = \int_A f\,d\mu$. Now, $D_\mu \eta$ can be calculated explicitely and is related to the weak derivative of $f$. (Actually I do not know what precisely is meant by $B_r^-(0)$ and hence, I could not work out the details.)