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For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due to many different definitions and theorems from probability books. I would appreiciate it if someone can clarify the terms and give a clear picture of the concepts. (Note that Did has answered some of my related questions before. Thank you, Did!)

  1. In Kallenberg's probability book, he defines weak convergence of a sequence measures to be

    Consider any probability measures $\mu$ and $\mu_1, \mu_2, \dots$ on some metric space $(S, \rho)$ with Borel a-field $S$, and say that $\mu_n$ converges weakly to $\mu$, if $\int f d\mu_n \to \int f d\mu$ for every $f \in C_b(S)$, the class of bounded, continuous functions $f: S \to \mathbb R$.

  2. Kallenberg defines vague convergence of a sequence of measures to be

    Consider the space $\mathcal M = \mathcal M(\mathbb R^d) $of locally finite mea- sures on $\mathbb R^d$. On $\mathcal M$ we may introduce the vague topology, generated by the mappings $\mu \mapsto \int f d\mu$ for all $f \in C_K^+$, the class of continuous functions $f: \mathbb R^d \to \mathbb R_+$ with compact support. In particular, $\mu_n$ is said to converge vaguely to $\mu$ if $\mu_n f \to \mu f$ for all $f \in C_K^+$. If the $\mu_n$ are probability measures, then clearly $\mu(\mathbb R^d) < 1$.

  3. Folland in his real ananlysis book defines vague topology and therefore vague convergence for complex Radon measures on a locally compact Hausdorff (LCH) space $X$ as weak* topology and weak* convergence wrt $C_0(X)$. He says the term "vague" is common in probability theory, and has the advantage of forming an adverb more gracefully than "weak*". The vague topology is sometimes called the weak topology, but this terminology conflicts with his, since $C_0(X)$ is rarely reflexive.
  4. In Kai Lai Chung's probability book, a sequence of subprobability measures $\mu_n$ on $\mathbb R$ are defined to vaguely converge to another subprobability measure $\mu$, if there exists a dense subset $D$ of $\mathbb R$ s.t. $\forall a, b \in D, a < b, \mu_n((a,b]) \to \mu((a,b])$.
  5. Next in Chung's, Theorem 4.4.1 says in case of subprobability measures, vague convergence is equivalent to weak* convergence wrt $C_0(\mathbb R)$ and $C_K(\mathbb R)$. Theorem 4.4.2 says in case of probability measures, vague convergence is equivalent to weak* convergence wrt $C_b(\mathbb R)$.

I was wondering if the above definitions of weak convergence and vague convergence are all weak* convergence, in the sense that the measures form (a subset of) the continuous dual of $C_b$, $C_K$, $C_K^+$, and $C_0$?

When defining vague convergence and vague topology, why does kallenberg use $C_K^+$ instead of $C_K$, Folland use $C_0$, and Kai Lai Chung uses $C_K$, $C_0$ and $C_b$? Are their definitions of vague convergence consistent with each other?

Among the convergences of measures wrt $C_b$, $C_K$, $C_K^+$, and $C_0$, when does which imply which? When is which equivalent to which?

The last question is to see if there are some unifications of the above concepts. Can the above definitions be generalized to more general measures (probability measures, subprobability measures, locally finite measures are used in the definitions above), and to more general underlying spaces (metric space, $\mathbb R^d$ and $\mathbb R$ are used in the definitions above)?

Thanks and regards!!

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There are a lot of questions here and I don't think I can answer them all, so I'll just leave this part as a comment. A critical fact for probability is that vague convergence of subprobability measures is what you get with $C_0$ or $C_K$ test functions (that is, compactly supported or vanishing at infinity) while weak convergence, which is what you get with $C_b$ test functions, is equivalent to vague convergence plus tightness. Intuitively, compactly supported functions can't tell if mass escapes to infinity, while bounded functions can. Example: $\delta_n \to 0$ vaguely, but not weakly. – Chris Janjigian Feb 25 '13 at 17:19
@ChrisJanjigian: Thanks! I am happy to learn that! I know there are many, but I think putting them together may help to reveal a clearer picture. – Tim Feb 25 '13 at 17:36
@ChrisJanjigian: (1) If I understand your comment correctly, for subprobability measures, weak convergence is equivalent to vague convergence plus tightness? is the measure space assumed to satisfy some other conditions? What is some reference for the statement? (2) Can the measures be generalized to be more general than subprobability measures? (3) If the subprobability measures happen to be probability measures, do we still need tightness for the equivalence? – Tim Feb 27 '13 at 17:15
(4) "A critical fact for probability is that vague convergence of subprobability measures is what you get with C0 or CK test functions". Are convergences wrt C0 and with CK still equivalent for more general measures than subprobability measures? – Tim Feb 27 '13 at 17:26
(1) actually I assumed that we are working on a Polish space when I said that. That is the usual assumption for the equivalence of weak convergence and vague convergence + tightness. You can find a proof of that result in Billingsley's Convergence of Probability Measures as theorem 5.2. (2) The condition of tightness is fairly specific to finite measures and you can always assume a finite measure with positive mass is a probability measure by rescaling, so I am not sure what you mean. (3) The same counterexample as above shows that you need tightness. – Chris Janjigian Feb 27 '13 at 17:53

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