# Tightness of measures

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In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."

Mathematics includes the study of such topics as quantity, structure, space, and change.

Infinity is a concept describing something without any bound, or something larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. This idea is also at the basis of infinitesimal calculus.

## Definitions

Let ${\displaystyle (X,T)}$ be a topological space, and let ${\displaystyle \Sigma }$ be a σ-algebra on ${\displaystyle X}$ that contains the topology ${\displaystyle T}$. (Thus, every open subset of ${\displaystyle X}$ is a measurable set and ${\displaystyle \Sigma }$ is at least as fine as the Borel σ-algebra on ${\displaystyle X}$.) Let ${\displaystyle M}$ be a collection of (possibly signed or complex) measures defined on ${\displaystyle \Sigma }$. The collection ${\displaystyle M}$ is called tight (or sometimes uniformly tight) if, for any ${\displaystyle \varepsilon >0}$, there is a compact subset ${\displaystyle K_{\varepsilon }}$ of ${\displaystyle X}$ such that, for all measures ${\displaystyle \mu \in M}$,

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

In mathematics, and more specifically in topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points ; however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open, or no set can be open but the space itself and the empty set.

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.

${\displaystyle |\mu |(X\setminus K_{\varepsilon })<\varepsilon .}$

where ${\displaystyle |\mu |}$ is the total variation measure of ${\displaystyle \mu }$. Very often, the measures in question are probability measures, so the last part can be written as

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire probability space.

${\displaystyle \mu (K_{\varepsilon })>1-\varepsilon .\,}$

If a tight collection ${\displaystyle M}$ consists of a single measure ${\displaystyle \mu }$, then (depending upon the author) ${\displaystyle \mu }$ may either be said to be a tight measure or to be an inner regular measure .

In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.

If ${\displaystyle Y}$ is an ${\displaystyle X}$-valued random variable whose probability distribution on ${\displaystyle X}$ is a tight measure then ${\displaystyle Y}$ is said to be a separable random variable or a Radon random variable.

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon. More specifically, a random variable is defined as a function that maps the outcomes of an unpredictable process to numerical quantities, typically real numbers. It is a variable, in the sense that it depends on the outcome of an underlying process providing the input to this function, and it is random in the sense that the underlying process is assumed to be random.

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. Examples of random phenomena can include the results of an experiment or survey.

## Examples

### Compact spaces

If ${\displaystyle X}$ is a metrisable compact space, then every collection of (possibly complex) measures on ${\displaystyle X}$ is tight. This is not necessarily so for non-metrisable compact spaces. If we take ${\displaystyle [0,\omega _{1}]}$ with its order topology, then there exists a measure ${\displaystyle \mu }$ on it that is not inner regular. Therefore, the singleton ${\displaystyle \{\mu \}}$ is not tight.

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

### Polish spaces

If ${\displaystyle X}$ is a compact Polish space, then every probability measure on ${\displaystyle X}$ is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on ${\displaystyle X}$ is tight if and only if it is precompact in the topology of weak convergence.

### A collection of point masses

Consider the real line ${\displaystyle \mathbb {R} }$ with its usual Borel topology. Let ${\displaystyle \delta _{x}}$ denote the Dirac measure, a unit mass at the point ${\displaystyle x}$ in ${\displaystyle \mathbb {R} }$. The collection

${\displaystyle M_{1}:=\{\delta _{n}|n\in \mathbb {N} \}}$

is not tight, since the compact subsets of ${\displaystyle \mathbb {R} }$ are precisely the closed and bounded subsets, and any such set, since it is bounded, has ${\displaystyle \delta _{n}}$-measure zero for large enough ${\displaystyle n}$. On the other hand, the collection

${\displaystyle M_{2}:=\{\delta _{1/n}|n\in \mathbb {N} \}}$

is tight: the compact interval ${\displaystyle [0,1]}$ will work as ${\displaystyle K_{\varepsilon }}$ for any ${\displaystyle \varepsilon >0}$. In general, a collection of Dirac delta measures on ${\displaystyle \mathbb {R} ^{n}}$ is tight if, and only if, the collection of their supports is bounded.

### A collection of Gaussian measures

Consider ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

${\displaystyle \Gamma =\{\gamma _{i}|i\in I\},}$

where the measure ${\displaystyle \gamma _{i}}$ has expected value (mean) ${\displaystyle m_{i}\in \mathbb {R} ^{n}}$ and covariance matrix ${\displaystyle C_{i}\in \mathbb {R} ^{n\times n}}$. Then the collection ${\displaystyle \Gamma }$ is tight if, and only if, the collections ${\displaystyle \{m_{i}|i\in I\}\subseteq \mathbb {R} ^{n}}$ and ${\displaystyle \{C_{i}|i\in I\}\subseteq \mathbb {R} ^{n\times n}}$ are both bounded.

## Tightness and convergence

Tightness is often a necessary criterion for proving the weak convergence of a sequence of probability measures, especially when the measure space has infinite dimension. See

## Exponential tightness

A strengthening of tightness is the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures ${\displaystyle (\mu _{\delta })_{\delta >0}}$ on a Hausdorff topological space ${\displaystyle X}$ is said to be exponentially tight if, for any ${\displaystyle \varepsilon >0}$, there is a compact subset ${\displaystyle K_{\varepsilon }}$ of ${\displaystyle X}$ such that

${\displaystyle \limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(X\setminus K_{\varepsilon })<-\varepsilon .}$

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## References

• 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.
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN   3-540-52013-9. MR 1102015 (See chapter 2)