Consider the set of finite positive measures on a complete, separable, but not compact, metric space $S$, endowed with the topology under which a sequence of finite positive measures $\{\mu_n\}$ converges to a finite positive measure $\mu$ if and only if $\int f d\mu_n \rightarrow \int f d\mu$ for every bounded continuous function $f$. If we restrict ourselves to the set of probability measures, i.e. measures $\mu$ such that $\mu(S)=1$, there is a metric (Prohorov's metric) that induces this topology. Is there a metric that induces this topology on the set of all finite, positive measures?
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$\begingroup$ What if you take $$|\mu-\nu|=\sup\left\{\,\left|\,\int\limits_S f\cdot(\mu-\nu)\,\right|\,\right\},$$ where the superemum is taken for all 1-Lipschitz function with supnorm at most 1? $\endgroup$– Anton PetruninCommented Dec 15, 2023 at 20:02
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$\begingroup$ You may assume that the metric is bounded above by $1$ define a Kantorovich (- Rubinshtein (or Wasserstein) metric like in mathoverflow.net/questions/120291/… $\endgroup$– Christophe LeuridanCommented Dec 17, 2023 at 16:45
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