Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are not required to be locally finite.
Let $ 1 \leq m \leq n $ be an integer. The upper density of a measure $ \phi $ at $ a \in \mathbf{R}^{n} $ is defined as
$ \Theta^{*m}(\phi,a) = \limsup_{r \to 0}\frac{ \phi(B(a,r))}{r^{m}}. $
My question is: is there a standard way to topologize $ B $ in order to have that $ \Theta^{*m} :B \times \mathbf{R}^{n} \rightarrow \mathbf{R} \cup \{\infty\} $ is a Borel map? Of course I am not looking for trivial topologies, but something that has reasonable properties (such as not too many open sets etc...).
In case I am looking for references about this problem.
Thank you for your answers.