Weak convergence of measures is convergence seen through bounded continuous test functions. It is the topology behind distributional convergence in probability and statistics.
Bounded-continuous test functions
Weak convergence tests measures by continuous observables.
Definition. Probability measures \(\mu_n\) on a metric space \(S\) converge weakly to \(\mu\), written \(\mu_n\Rightarrow\mu\), if \(\int f\,\mathrm{d}\mu_n\to\int f\,\mathrm{d}\mu\) for every bounded continuous \(f:S\to\mathbb R\).
Remark. Bounded continuous functions determine probability measures on metric spaces. Indeed, regularity approximates closed-set indicators from above by \[ f_k(x)=\max\{1-kd(x,F),0\}, \] so equality of all bounded-continuous integrals forces equality on closed sets and hence on the Borel sigma-algebra.
Portmanteau theorem
Weak convergence has many equivalent formulations.
Theorem (portmanteau). For probability measures on a metric space, the following are equivalent: \[ \begin{array}{ll} \text{(i)}&\mu_n\Rightarrow\mu;\\ \text{(ii)}&\limsup_n\mu_n(F)\le\mu(F)\quad\text{for every closed }F;\\ \text{(iii)}&\liminf_n\mu_n(G)\ge\mu(G)\quad\text{for every open }G;\\ \text{(iv)}&\mu_n(A)\to\mu(A)\quad\text{whenever }\mu(\partial A)=0;\\ \text{(v)}&\liminf_n\int f\,\mathrm{d}\mu_n\ge\int f\,\mathrm{d}\mu \quad\text{for every bounded lower-semicontinuous }f. \end{array} \]
Proof. Assume (i). For closed \(F\), the continuous functions \[ f_k(x)=\max\{1-kd(x,F),0\} \] decrease to \(\mathbf1_F\). Hence \[ \limsup_n\mu_n(F) \le\lim_n\int f_k\,\mathrm{d}\mu_n =\int f_k\,\mathrm{d}\mu, \] and \(k\to\infty\) gives (ii). Complements make (ii) and (iii) equivalent. If \(\mu(\partial A)=0\), then \[ \mu(A^\circ)\le\liminf_n\mu_n(A) \le\limsup_n\mu_n(A)\le\mu(\overline A), \] and both endpoint measures equal \(\mu(A)\), proving (iv).
To derive (v) from (iii), first take \(0\le f\le M\) lower semicontinuous. Its superlevel sets \(\{f>t\}\) are open, so the layer-cake formula and Fatou give \[ \liminf_n\int f\,\mathrm{d}\mu_n =\liminf_n\int_0^M\mu_n(f>t)\,\mathrm{d}t \ge\int_0^M\mu(f>t)\,\mathrm{d}t. \] Shift a bounded \(f\) by a constant. Finally, (v) applied to both \(f\) and \(-f\) for bounded continuous \(f\) gives (i).
Theorem (bounded-Lipschitz determination). On a separable metric space, \[ \mu_n\Rightarrow\mu \quad\Longleftrightarrow\quad \int f\,\mathrm{d}\mu_n\to\int f\,\mathrm{d}\mu \] for every \(f\) satisfying \(\|f\|_\infty\le1\) and \(\operatorname{Lip}(f)\le1\). The supremum of these integral differences defines the bounded-Lipschitz metric, which metrizes weak convergence.
Proof outline. One direction is immediate because bounded Lipschitz functions are bounded continuous. Conversely, approximate a bounded continuous function uniformly on a large compact set by Lipschitz infimal and supremal convolutions. Tightness of the single limit measure, together with the portmanteau inequalities, controls the complement. Separability supplies a countable determining subclass and turns the supremum into a metric.
Continuous mapping theorem
Weak convergence is preserved by continuous transformations.
Theorem (continuous mapping). If \(\mu_n\Rightarrow\mu\) on \(S\) and \(T:S\to E\) is measurable with discontinuity set \(D_T\) satisfying \(\mu(D_T)=0\), then \[ \mu_n\circ T^{-1}\Rightarrow\mu\circ T^{-1}. \] In particular, global continuity of \(T\) is sufficient.
Proof. For a closed \(F\subset E\), the set \(T^{-1}(F)\) need not be closed, but \[ \overline{T^{-1}(F)} \subset T^{-1}(F)\cup D_T. \] Portmanteau therefore gives \[ \limsup_n(\mu_n\circ T^{-1})(F) \le\mu(\overline{T^{-1}(F)}) \le\mu(T^{-1}(F)), \] because \(\mu(D_T)=0\). Another application of portmanteau proves weak convergence of the pushforwards.
Convergence on the real line
Distribution functions provide a one-dimensional criterion.
Theorem. Probability measures \(\mu_n\) on \(\mathbb R\) converge weakly to \(\mu\) if and only if \(F_n(x)\to F(x)\) at every continuity point of \(F\).
Proof. The forward direction follows from portmanteau applied to \((-\infty,x]\) and \((-\infty,x)\). For the reverse direction, approximate bounded continuous functions by step functions whose discontinuity points are continuity points of \(F\), controlling the tails by choosing large compact intervals.
Weak convergence does not control unbounded moments
Example. Let \[ \mu_n=\left(1-\frac1n\right)\delta_0+\frac1n\delta_n. \] For every bounded continuous \(f\), \[ \int f\,\mathrm{d}\mu_n =\left(1-\frac1n\right)f(0)+\frac1nf(n)\longrightarrow f(0), \] so \(\mu_n\Rightarrow\delta_0\). Nevertheless, \[ \int x\,\mathrm{d}\mu_n(x)=1 \] for every \(n\), while the limiting mean is \(0\); second moments even diverge. Convergence of moments requires uniform integrability of the corresponding powers.
Remark. Central limit theorems, bootstrap consistency, empirical-process convergence, and posterior weak convergence all use this topology or its function-space analogues. The recurring proof pattern is tightness plus identification of every subsequential limit.