Perfect measure

A concept introduced by B.V. Gnedenko and A.N. Kolmogorov in [1] with the aim of "attaining a full harmony between abstract measure theory and measure theory in metric spaces" . The subsequent development of the theory has revealed other aspects of the value of this concept. On the one hand the class of perfect measures is very wide, and on the other, a number of unpleasant technical complications that occur in general measure theory do not arise if one restricts to perfect measures.

A finite measure on a -algebra of subsets of a set is called perfect if for any real-valued measurable function on and any set such that ,

where is the class of open subsets of . For to be perfect, it is necessary that for any real-valued measurable function on there exists a Borel set such that , and sufficient that for any real-valued measurable function on and any set for which there exists a Borel set such that

Every discrete measure is perfect. A measure defined on a -algebra of subsets of a separable metric space that contains all open sets is perfect if and only if the measure of any measurable set is the least upper bound of the measures of its compact subsets. The restriction of a perfect measure defined on to any -subalgebra of is perfect. A measure induced by a perfect measure on any subset with is perfect. The image of a perfect measure under a measurable mapping of into another measurable space is perfect. A measure is perfect if and only if its completion is perfect. For every measure on any -subalgebra of a -algebra of subsets of a set to be perfect it is necessary and sufficient that for any real-valued -measurable function the set is universally measurable (that is, it belongs to the domain of definition of the completion of every Borel measure on ). If and if is the -algebra of Borel subsets of , then every measure on is perfect if and only if is universally measurable.

Every space with a perfect measure such that has a countable numbers of generators separating points of (that is, for all , , there is an : , or , ) is almost isomorphic to some space , consisting of the Lebesgue measure on a finite interval and of a countable sequence (possibly empty) of points of positive mass (i.e., there is an with and a one-to-one mapping of onto such that and are measurable and ).

Let be any index set and let be a given space with a perfect measure for each . Put and let be the algebra generated by the class of sets of the form . If is a finitely-additive measure on such that for all and , then: 1) is countably additive on ; and 2) the extension of to the -algebra generated by is perfect.

Let be a space with a perfect probability measure and let , be two -subalgebras of the -algebra , where has a countable number of generators. Then there is a regular conditional probability on given , i.e. there is a function on such that: 1) for a fixed , is a probability measure on ; 2) for a fixed , is measurable with respect to ; and 3) for all and . Moreover, the function can be chosen in such a way that the measures are perfect. Let , be two measurable spaces and let be a transition probability on , that is, is measurable with respect to and is a probability measure on for all , . If the are discrete and is a perfect probability measure on , then the measure is perfect.

Perfect measures are closely connected with compact measures. A class of subsets of is called compact if , and implies that for some . A finite measure on is called compact if there is a compact class such that for all and one can choose a and an such that and . Every compact measure is perfect. For a measure to be perfect it is necessary and sufficient that its restriction to any -subalgebra with a countable number of generators be compact.

References

[1]	B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)
[2]	E. Marczewski, "On compact measures" Fund. Math. , 40 (1953) pp. 113–124
[3]	C. Ryll-Nardzewski, "On quasi-compact measures" Fund. Math. , 40 (1953) pp. 125–130
[4]	V.V. Sazonov, "On perfect measures" Transl. Amer. Math. Soc. (2) , 48 (1965) pp. 229–254 Izv. Akad. Nauk SSSR Ser. Mat. , 26 (1962) pp. 391–414
[5]	D. Ramachandran, "Perfect measures" , 1–2 , Macmillan (1979)

References

[a1]	Kia-An Yen, "Forme mesurable de la théorie des ensembles sousliniens, applications à la théorie de la mesure" Scientia Sinica , XVIII (1975) pp. 444–463