 Convergence of measures

In mathematics, more specifically measure theory, there are various notions of the convergence of measures. Three of the most common notions of convergence are described below.
Contents
Total variation convergence of measures
This is the strongest notion of convergence shown on this page and is defined a follows. Let be a measurable space. The total variation distance between two (positive) measures μ and ν is then given by
If μ and ν are both probability measures, then the total variation distance is also given by
The equivalence between these two definitions can be seen as a particular case of the MongeKantorovich duality. From the two definitions above, it is clear that the total variation distance between probability measures is always between 0 and 2.
To illustrate the meaning of the total variation distance, consider the following thought experiment. Assume that we are given two probability measures μ and ν, as well as a random variable X. We know that X has law either μ or ν, but we do not know which one of the two. Assume now that we are given one single sample distributed according to the law of X and that we are then asked to guess which one of the two distributions describes that law. The quantity
then provides a sharp upper bound on the probability that our guess is correct.
Strong convergence of measures
For a measurable space, a sequence μ_{n} is said to converge strongly to a limit μ if
for every set in .
For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ_{n} of measures on the interval [1,1] given by converges strongly to Lebesgue measure, but it does not converge in total variation.
Weak convergence of measures
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 σalgebra Σ. We say that a sequence of probability measures on (S, Σ), P_{n}, n = 1, 2, ..., converges weakly to the probability measure P, and write
if any of the following equivalent conditions is true:
 E_{n}ƒ → Eƒ for all bounded, continuous functions ƒ;
 E_{n}ƒ → Eƒ for all bounded and Lipschitz functions ƒ;
 limsup E_{n}ƒ ≤ Eƒ for every upper semicontinuous function ƒ bounded from above;
 liminf E_{n}ƒ ≥ Eƒ for every lower semicontinuous function ƒ 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 case S = R with its usual topology, if F_{n}, F denote the cumulative distribution functions of the measures P_{n}, P respectively, then P_{n} converges weakly to P if and only if lim_{n→∞} F_{n}(x) = F(x) for all points x ∈ R at which F is continuous.
For example, the sequence where P_{n} is the Dirac measure located at 1/n converges weakly to the Dirac measure located at 0 (if we view these as measures on R with the usual topology), but it does not converge strongly. This is intuitively clear: we only know that 1/n is "close" to 0 because of the topology of R.
This definition of weak convergence can be extended for S any metrizable topological space. It also defines a weak topology on P(S), the set of all probability measures defined on (S, Σ). The weak topology is generated by the following basis of open sets:
where
If S is also separable, then P(S) is metrizable and separable, for example by the Lévy–Prokhorov metric, if S is also compact or Polish, so is P(S).
If S is separable, it naturally embeds into P(S) as the (closed) set of dirac measures, and its convex hull is dense.
There are many "arrow notations" for this kind of convergence: the most frequently used are , and .
Weak convergence of random variables
Let be a probability space and X be a metric space. If X_{n}, X: Ω → X is a sequence of random variables then X_{n} is said to converge weakly (or in distribution or in law) to X as n → ∞ if the sequence of pushforward measures (X_{n})_{∗}(P) converges weakly to X_{∗}(P) in the sense of weak convergence of measures on X, as defined above.
References
 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 3764324287.
 Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc.. ISBN 0471007102.
 Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc.. ISBN 0471197459.
See also
 Convergence of random variables
 Prokhorov's theorem
 Lévy–Prokhorov metric
 Tightness of measures
Categories: Measure theory
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