Borel measure

of sets

2020 Mathematics Subject Classification: Primary: 28C15 [MSN][ZBL]

A non-negative function of the subsets of a topological space possessing the following properties: 1) its domain of definition is the -algebra of Borel sets (cf. Borel set) in , i.e. the smallest class of subsets in containing the open sets and closed with respect to the set-theoretic operations performed a countable number of times; and 2) if when , i.e. is countably additive. A Borel measure is called regular if

where belongs to the class of closed subsets in . The study of Borel measures is often connected with that of Baire measures, which differ from Borel measures only in their domain of definition: They are defined on the smallest -algebra with respect to which all continuous functions on are measurable. A Borel measure (or a Baire measure ) is said to be -smooth if for any net of closed sets which satisfies the condition (or for any net of sets which are zero sets of continuous functions and such that ). A Borel measure (or Baire measure ) is said to be tight if

where is the class of compact subsets on (or

where

Tightness and -smoothness are restrictions which ensure additional smoothness of measures, and which in fact often hold. Under certain conditions Baire measures can be extended to Borel measures. For instance, if is a completely-regular Hausdorff space, then any -smooth (tight) finite Baire measure can be extended to a regular -smooth (tight) finite Borel measure. In the study of measures on locally compact spaces Borel measures (or Baire measures) is the name sometimes given to measures defined on the -ring of sets generated by the compact (or -compact) sets and which are finite on compact sets. Often, by the Borel measure on the real line one understands the measure defined on the Borel sets such that its value on an arbitrary segment is equal to the length of that segment.

References

[1]	V.S. Varadarajan, "Measures on topological spaces" Transl. Amer. Math. Soc. Ser. 2 , 48 (1965) pp. 161–228 Mat. Sb. , 55 (97) : 1 (1961) pp. 35–100 MR0148838
[2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[3]	J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) MR0272004 Zbl 0203.49901

References

[a1]	H.L. Royden, "Real analysis", Macmillan (1968)
[a2]	A.C. Zaanen, "Integration" , North-Holland (1967) MR0222234 Zbl 0175.05002
[a3]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) MR0055409 Zbl 0052.05301
[a4]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 MR0210528 Zbl 0142.01701
[a5]	A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) MR0178100 Zbl 0135.11301
[a6]	C.D. Aliprantz, O. Burleinshaw, "Principles of real analysis" , North-Holland (1981)