Borel measure

of sets

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

Definition

The terminology Borel measure is used by different authors with different meanings:

(A) Some authors use it for measures $\mu$ on the $\sigma$-algebra $\mathcal{B}$ of Borel subsets of a given topological space $X$, i.e. functions $\mu:\mathcal{B}\to [0, \infty]$ which are countably additive.

(B) Some authors use it for measures $\mu$ on the $\sigma$-algebra of Borel sets of a locally compact topological space satisfying the additional property that $\mu (K)<\infty$ for every compact set $K$ (see for instance Section 52 of [Ha] or Section 14 of Chapter 1 of [Ro]).

(C) Some authors (cp. with Definition 1.5(b) of [Ma] or with Section 1.1 of [EG]) use it for outer measures $\mu$ on a topological space $X$ for which the Borel sets are $\mu$-measurable (hence the difference between acception (A) and (B) is small, however when coming to the terminology Borel regular we will see more important discrepancies).

Borel regular measures

In these three different contexts Borel regular measures are then defined as follows:

(A) Borel measures $\mu$ for which \[ \sup\; \{\mu (C):C\subset E\mbox{ is closed}\} = \mu (E) \qquad \mbox{for any Borel set '"`UNIQ-MathJax15-QINU`"'.} \]

(B) Borel measures $\mu$ such that \[ \sup\; \{\mu (C):C\subset E\mbox{ is compact}\} = \mu (E) \qquad \mbox{for any Borel set '"`UNIQ-MathJax17-QINU`"'} \] and \[ \inf\; \{\mu (U):U\supset E\mbox{ is open}\} = \mu (E) \qquad \mbox{for any Borel set '"`UNIQ-MathJax18-QINU`"'} \] (cp. with Chapter 52 of [Ha]).

(C) Borel (outer) measures such that for any $A\subset X$ there is a Borel set $B$ with $\mu(A)=\mu(B)$ (cp. with Definition 1.5(3) of [Ma]).

Warning: The authors using terminology (A) call tight the measures called Borel regular by authors using terminology (B). Moreover they call $\tau$-smooth those Borel measures for which $\mu (F_k)\to 0$ for any sequence (or, more in general, net) of closed sets with $F_k\downarrow \emptyset$.

Warning: The authors using terminology (C) use the term Radon measures for the measures which the authors as in (B) call Borel regular and the authors in (A) call tight (cp. with Definition 1.5(4) of [Ma]).

Comments

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 $\sigma$-algebra $\mathcal{B}_0$ for which continuous functions are $\mathcal{B}_0$ measurable (cp. with Sections 51 and 52 of [Ha]). In particular observe that the two concepts coincide on topological spaces such that for any open set $U$ there is a continuous function $f$ with $f^{-1} (]0, \infty[) = U$ (cp. with Separation axiom).

The concept of tightness and $\tau$-smoothness can be extended to Baire measures as well (and in fact the authors using terminology (B) call Baire regular the Baire measures which are tight, cp. with Section 52 of [Ha]).

If $X$ is a completely regular space, then any $\tau$-smooth (tight) finite Baire measure can be extended to a regular $\tau$-smooth (tight) finite Borel measure (cp. with Theorem D of Section 54 in [Ha]).

References

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[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Ha]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ma]	P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[N]	J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970) MR0272004 Zbl 0203.49901
[V]	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
[Ro]	H.L. Royden, "Real analysis", Macmillan (1968) MR0151555 Zbl 0197.03501
[R]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) MR0055409 Zbl 0052.05301
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[Z]	A.C. Zaanen, "Integration" , North-Holland (1967) MR0222234 Zbl 0175.05002