Weak Convergence

A sequence of probability measures (μ_n) on a metric space converges weakly to μ if ∫ f dμ_n → ∫ f dμ for every bounded continuous function f; equivalently, distribution functions converge at continuity points of the limit.

Mode of convergence in a topological vector space (typically a Banach or Hilbert space, or space of measures) where a sequence x_n converges weakly to x if every continuous linear functional φ satisfies φ(x_n) → φ(x); in measure spaces this generalizes weak-* or narrow convergence of measures.