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In mathematics, there are various notions of the convergence of measures in measure theory. Broadly speaking, there are two kinds of convergence, strong convergence and weak convergence.

- $rho\; (mu,\; nu)\; :=\; sup\; left\{\; left.\; int\_\{X\}\; f(x)\; ,\; mathrm\{d\}\; (mu\; -\; nu)\; (x)\; right|\; mathrm\{continuous,\}\; f\; :\; X\; to\; [-1,\; 1]\; subset\; mathbb\{R\}\; right\}$

- $tau\; (mu,\; nu)\; :=\; sup\; left\{\; left.\; |\; mu\; (A)\; -\; nu\; (A)\; |\; right|\; A\; in\; mathcal\{F\}\; right\}.$

There are (at least) five definitions of weak convergence of a sequence of measures, some of which are more general than others. The following equivalence result is sometimes known as the portmanteau theorem, and shows the equivalence of four such definitions for probability measures on a general metrizable topological space, and a fifth condition, which makes sense only for distributions on the real line.

Let (Ω, T) be a topological space with its Borel σ-algebra Borel(Ω), and let P(Ω) denote the collection of all probability measures defined on (Ω, Borel(Ω)). Consider here the case of metrizable Ω (we need some restriction of the topology and second countable is not sufficient). If Ω is also separable, P(Ω) (with the weak topology defined below) is metrizable, for example by the Lévy-Prokhorov metric. Let μ_{n}, n = 1, 2, ..., be a sequence in P(Ω) and let μ ∈ P(Ω). Then the following conditions are all equivalent:

- $lim\_\{n\; to\; infty\}\; int\_\{Omega\}\; f\; ,\; mathrm\{d\}\; mu\_\{n\}\; =\; int\_\{Omega\}\; f\; ,\; mathrm\{d\}\; mu$ for all bounded and continuous functions f : Ω → R (sometimes referred to as "test functions");
- limsup
_{n→∞}μ_{n}(C) ≤ μ(C) for all closed subsets C of Ω; - liminf
_{n→∞}μ_{n}(U) ≥ μ(U) for all open subsets U of Ω; - lim
_{n→∞}μ_{n}(A) = μ(A) for all so-called "μ-continuity" subsets A of Ω: those sets A with μ(∂A) = 0, where ∂A denotes the boundary of A; - in the case Ω = R with its usual topology, if F
_{n}, F denote the cumulative distribution functions of the measures μ_{n}, μ respectively, then lim_{n→∞}F_{n}(x) = F(x) for all points x ∈ R at which F is continuous.

There are many "arrow notations" for this kind of convergence: the most frequently used are $mu\_\{n\}\; Rightarrow\; mu$, $mu\_\{n\}\; rightharpoonup\; mu$ and $mu\_\{n\}\; xrightarrow\{mathcal\{D\}\}\; mu.$.

- Ambrosio, L., Gigli, N. & Savaré, G. (2005).
*Gradient Flows in Metric Spaces and in the Space of Probability Measures*. Basel: ETH Zürich, Birkhäuser Verlag. ISBN 3-7643-2428-7. - Billingsley, Patrick (1995).
*Probability and Measure*. New York, NY: John Wiley & Sons, Inc.. ISBN 0-471-00710-2. - Billingsley, Patrick (1999).
*Convergence of Probability Measures*. New York, NY: John Wiley & Sons, Inc.. ISBN 0-471-19745-9.

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Last updated on Thursday August 07, 2008 at 15:31:30 PDT (GMT -0700)

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