weak convergence of probability measures
Volume I told you that a sequence of random variables X_n converges in distribution to X when their cumulative distribution functions converge at every continuity point of the limit. That definition is tied to the real line, where a CDF makes sense. But limit theorems live on richer spaces: random functions live in C[0,1], random point patterns live in spaces of measures, random graphs in their own state spaces. Weak convergence is the right generalization of 'convergence in distribution' to any metric space S, and it is the central organizing concept of advanced limit theory.
Let S be a metric space with its Borel sigma-algebra, and let P(S) be the set of Borel probability measures on S. A sequence mu_n in P(S) converges weakly to mu, written mu_n => mu, if E_{mu_n}[f] -> E_mu[f] for every bounded continuous function f: S -> R, that is, integral f dmu_n -> integral f dmu for all f in C_b(S). The word 'weak' is borrowed from functional analysis: we are testing the measures only against the test functions C_b(S), so this is convergence in the weak-star sense, viewing measures as a subset of the dual of C_b(S). For random variables X_n with laws mu_n, X_n => X means E[f(X_n)] -> E[f(X)] for all bounded continuous f.
Why bounded continuous and not, say, all bounded measurable functions? Because requiring convergence against indicators of arbitrary sets is far too strong (it would force the measures to be eventually equal in a discrete sense), while continuous test functions are insensitive to mass sitting exactly on a boundary, which is exactly what allows a sequence of atoms to converge to a smooth limit, or a discrete approximation to converge to a continuum object. The whole edifice of this field, the Portmanteau theorem, Prokhorov's theorem, the Skorokhod representation, exists to give you many equivalent and workable handles on this one definition.
Let mu_n be the uniform measure on the n points {k/n : k = 1, ..., n} in [0,1]. For any continuous f, integral f dmu_n = (1/n) sum_{k=1}^n f(k/n), which is a Riemann sum converging to integral_0^1 f(x) dx. So mu_n => Lebesgue measure on [0,1], even though no single mu_n is anywhere near continuous and the densities never converge.
A discrete approximation converging weakly to a continuous limit: the hallmark of why 'weak' is the right notion.
Weak convergence does NOT imply convergence of means or any other moments. If mu_n puts mass 1 - 1/n at 0 and 1/n at n, then mu_n => delta_0 but the mean stays 1; you need uniform integrability to pass to moments.