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.
Best Answer
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.