Convergence of measures

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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.

[edit] Strong convergence of measures

Let $(X, \mathcal{F})$ be a measure space. If the collection of all measures (or, frequently, just probability measures) on $(X, \mathcal{F})$ can be given some kind of metric, then convergence in this metric is usually referred to as strong convergence. Examples include the Radon metric

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

and the total variation metric

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

[edit] Weak convergence of measures

It has been suggested that Weak convergence of measures be merged into this article or section. (Discuss)

If the convergence of a sequence of measures $(\mu_{n})_{n = 1}^{\infty}$ to another measure $μ$ is determined by a pointwise duality with respect to a class of "nice" test functions, then this is usually referred to as weak convergence. For example, the usual definition of weak convergence of measures on $X$ is that, for each bounded and continuous $f : X \to \mathbb{R}$ ,

$\int_{X} f(x) \, \mathrm{d} \mu_{n} (x) \to \int_{X} f(x) \, \mathrm{d} \mu (x)$ as $n \to \infty.$

Generally, strong convergence implies weak convergence, but not the other way around. (Hence the appellations "strong" and "weak".) It is possible to construct metrics whose topology agrees with the topology of weak convergence: an example is the Lévy-Prokhorov metric.