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Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure.

Notation used throughout:-

$X=\mathbb{R}^n$

$\mathcal{P}(X)$- Space of Borel probability measures on X

$C_b(X)$- Space of continuous and bounded functions on X

Definitions:-

Narrow Convergence: A sequence $(\mu_n)\subset\mathcal{P}(X)$ is narrowly convergent to $\mu\in\mathcal{P}(X)$ if $\int_Xfd\mu_n\xrightarrow{n\rightarrow \infty}\int_Xfd\mu$ for every $f\in C_b(X)$.

Tightness: A set $\mathcal{K}\subset\mathcal{P}(X)$ is tight if $\forall \epsilon>0 \ \ , \exists K_\epsilon \text{ compact in $X$ such that } \mu(X\backslash K_\epsilon)\leq \epsilon \ \ \forall \mu\in\mathcal{K}$.

Prokhorov's theorem: If a set $\mathcal{K}\subset \mathcal{P}(X)$ is tight then $\mathcal{K}$ is relatively compact in $\mathcal{P}(X)$.

Question: Does there exists a generalization of Prokhorov's theorem to $\mathcal{M}(X)$, the space of finitely additive signed measures on X? Any references would be welcome.

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  • $\begingroup$ Quick Motivating Idea: Bounded Lipschitz (dual) metric metrizes the narrow topology on the space of finitely additive signed measures. So can we have a similar characterization (like Prokhorov’s theorem) of compact sets in this space? $\endgroup$
    – UPS
    Aug 29, 2013 at 19:59
  • $\begingroup$ Just for reference, Prokhorov's theorem can be extended to complex Borel measures. Please see here. $\endgroup$
    – Analyst
    Nov 8, 2022 at 10:59

1 Answer 1

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There is a nice generalization of the Prokhorov's theorem to the space of Borel measures. For proof see Bogachev's Measure Theroy Vol 2 (Thm 8.6.2).

Let $X$ be a complete seperable metric space and $\mathcal{M}$ a family of Borel measures on $X$. Then the following statements are equivalent:

(1) Every sequence $\mu_n\subset\mathcal{M}$ contains a weakly convergent susequence.

(2) The family $\mathcal{M}$ is tight and uniformly bounded in the total variation norm.

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