In several articles available over the internet, it is written that:
.. $M_{loc}(\Omega)$ is the space of locally finite Borel Measures on $\Omega$ with the standard locally convex topology generated by semi-norms $p_{\Phi}(\mu) = Var(\Phi_{\mu})$ where: $\Phi = \Phi (x) \in C_{0}(\Omega)$ .
But, nowhere I am able to find the definition of this "Var." As it seems to me, it may be something like "Total Variation of a Measure" ; but I couldn't find the exact definition of the above "Var."
Can someone please give me the precise definition of "$Var (\Phi_{\mu})$" for the above case???
Thank You!
The only thing which makes sense to me is that $\Phi_\mu$is the finite measure given by $d\phi_\mu = \Phi \,d\mu$.
Then $Var(\Phi_\mu)$ is indeed the total variation of $\Phi_\mu$, which is the same as the norm of $\Phi_\mu$ as a linear functional on $C_b (\Omega)$, i.e. $$ Var(\Phi_\mu) = \sup_{\|f\|\leq 1, f\in C(\Omega)} |\int f\Phi \,d\mu|. $$