A mode of convergence for probability measures on a metric (or topological) space: a sequence of probability measures μ_n converges weakly to μ if ∫ f dμ_n → ∫ f dμ for every bounded continuous test function f (equivalently via distribution functions at continuity points or Portmanteau conditions).

Principle

Weak convergence is the convergence of expectations against bounded continuous observables; it captures convergence of macroscopic distributions while ignoring small-scale oscillations invisible to continuous tests.

Demonstration

Let μ_n be the uniform distribution on [0,1/n]. For any bounded continuous f on R, ∫ f dμ_n → f(0), so μ_n converges weakly to the Dirac measure at 0.

Misapplication

Assuming weak convergence implies convergence of densities in L^1 or convergence of probabilities of boundaries of sets; weak convergence does not control total variation or unbounded test functions without extra moment conditions.

Consequence

Weak convergence yields convergence of integrals for bounded continuous functions and is sufficient for many limit theorems (e.g., central limit theorem statements in distribution); it allows compactness via Prokhorov's theorem under tightness.

Reversal

Strong (total variation) convergence implies weak convergence but not conversely; sequences can converge weakly while retaining substantial mass oscillations at small scales or escaping in ways invisible to bounded continuous tests.

Boundary

Typically defined for Borel probability measures on metric spaces; differs from vague convergence (which tests only against compactly supported continuous functions) and requires tightness for sequential compactness in noncompact spaces.

Semantic Tension

Tension exists between weak convergence and modes like total variation or convergence of moments: weak convergence is weaker and more topological, while stronger norms give quantitative control but require more assumptions.

Synthesis

Weak convergence of probability measures means convergence of expectations against bounded continuous observables, a topological notion of distributional convergence that captures macroscopic limiting behavior without requiring strong norm control.