Tightness of measures
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity". Definitions. Let formula_1 be a Hausdorff space, and let formula_2 be a σ-algebra on formula_3 that contains the topology formula_4. (Thus, every open subset of formula_3 is a measurable set and formula_2 is at least as fine as the Borel σ-algebra on formula_3.) Let formula_8 be a collection of (possibly signed or complex) measures defined on formula_2. The collection formula_8 is called tight (or sometimes uniformly tight) if, for any formula_11 , there is a compact subset formula_12 of formula_3 such that, for all measures formula_14, where formula_16 is the total variation measure of formula_17. Very often, the measures in question are probability measures, so the last part can be written as If a tight collection formula_8 consists of a single measure formula_17, then (depending upon the author) formula_17 may either be said to be a tight measure or to be an inner regular measure. If formula_22 is an formula_3-valued random variable whose probability distribution on formula_3 is a tight measure then formula_22 is said to be a separable random variable  or a Radon random variable. Examples. Compact spaces. If formula_3 is a metrisable compact space, then every collection of (possibly complex) measures on formula_3 is tight. This is not necessarily so for non-metrisable compact spaces. If we take formula_28 with its order topology, then there exists a measure formula_17 on it that is not inner regular. Therefore, the singleton formula_30 is not tight. Polish spaces. If formula_3 is a compact Polish space, then every probability measure on formula_3 is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on formula_3 is tight if and only if it  is precompact in the topology of weak convergence. A collection of point masses. Consider the real line formula_34 with its usual Borel topology. Let formula_35 denote the Dirac measure, a unit mass at the point formula_36 in formula_34. The collection is not tight, since the compact subsets of formula_34 are precisely the closed and bounded subsets, and any such set, since it is bounded, has formula_40-measure zero for large enough formula_41. On the other hand, the collection is tight: the compact interval formula_43 will work as formula_12 for any formula_11. In general, a collection of Dirac delta measures on  formula_46 is tight if, and only if, the collection of their supports is bounded. A collection of Gaussian measures. Consider formula_41-dimensional Euclidean space formula_46 with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures where the measure formula_50 has expected value (mean) formula_51 and covariance matrix formula_52. Then the collection formula_53 is tight if, and only if, the collections formula_54 and formula_55 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 formula_56 on a Hausdorff topological space formula_3 is said to be exponentially tight if, for any formula_11, there is a compact subset formula_12 of formula_3 such that