In mathematics and statistics, weak convergence (also known as narrow convergence or weak-* convergence, which is a more appropriate name from the point of view of functional analysis, but less frequently used) is one of many types of convergence relating to the convergence of measures. It depends on a topology on the underlying space and thus is not a purely measure theoretic notion.
There are several equivalent definitions of weak convergence of a sequence of measures, some of which are (apparently) more general than others. The equivalence of these conditions is sometimes known as the portmanteau theorem.
Definition. Let $S$ be a metric space with its Borel σ-algebra $Σ$. We say that a sequence of probability measures $P_n$ on $(S, Σ), n = 1, 2, ...,$ converges weakly to the probability measure $P$, if any of the following equivalent conditions is true (here $E_n$ denotes expectation with respect to $P_n$ while $E$ denotes expectation with respect to $P$):
- $E_nf → Ef$ for all bounded, continuous functions $f$;
- $E_nf → Ef$ for all bounded and Lipschitz functions $f$;
- $\limsup E_nf ≤ Ef$ for every upper semi-continuous function $f$ bounded from above;
- $\liminf E_nf ≥ Ef$ for every lower semi-continuous function $f$ bounded from below;
- $\limsup P_n(C) ≤ P(C)$ for all closed sets $C$ of space $S$;
- $\liminf P_n(U) ≥ P(U)$ for all open sets $U$ of space $S$;
- $\lim P_n(A) = P(A)$ for all continuity sets $A$ of measure $P$.
In the page you quote, Wikipedia defines weak convergence of (probability) measures. This mode of convergence should be called weak* rather than weak because it refers to the convergence against any bounded continuous functions, and the space $M_1$ of probability measures is included in the dual of the space $C_b$ of bounded continuous functions. That is, $(\mu_n)$ in $M_1$ converges to $\mu$ in $M_1$ if and only if $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ for every $f$ in $C_b$ and this is a weak* convergence because $M_1\subset(C_b)^*$.
Now I wonder why this convergence mode is called weak? Can "weak" be explained from the point of view of functional analysis, or is it explained from some other point of view?
Thanks and regards!