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Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ with compact support equipped with the locally convex inductive limit topology. Equip $M$ with the vague topology (i.e. the weak-* topology). Then $M$ is not first-countable and thus not metrizable. However, the subspace of positive measures is Polish.

(i) Is $M$ Suslin or even Lusin?

(ii) Is the Borel $\sigma$-algebra on $M$ the same as the $\sigma$-algebra generated by the evaluations $M \to \mathbb{R}$, $\mu \mapsto \mu B$ for bounded Borel-measurable sets $B \subseteq \mathbb{R}$?

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  • $\begingroup$ Why is the subset of positive measures polish? I believe that the set of probability measures is polish. $\endgroup$
    – Jochen Wengenroth
    Commented Mar 19, 2015 at 15:25
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    $\begingroup$ Did you try something with the Jordan-Hahn decomposition? For real-valued measures you would have a continuous surjection $M_+\times M_+ \to M$, $(\nu,\mu)\mapsto \nu-\mu$. $\endgroup$
    – Jochen Wengenroth
    Commented Mar 19, 2015 at 15:27
  • $\begingroup$ see Kallenberg, Random Measures, Appendix 15.7.7. He also refers to Bourbaki. The idea is to take a countable base $(C_k)_k$ of $\mathbb{R}$, approximate the indicator of $C_k$ by a sequence of functions $f_{ki}$ in $C_c$ and take $(f_{ki})_{k,i}$ as the countable set enumerated by $j$ to define a metric on $M$ by something like $\rho(\mu, \mu') := \sum_{j=1}^\infty \frac{1}{2^j} (1 - e^{-|\mu f_j - \mu' f_j|})$. $\endgroup$
    – yada
    Commented Mar 19, 2015 at 15:33
  • $\begingroup$ Ok, Jordan-Hahn decomposition seems to be (sequentially) continuous (but clearly not injective). Thus, $M$ is at least Suslin and separable. $\endgroup$
    – yada
    Commented Mar 19, 2015 at 16:19

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