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''of sets''
 
''of sets''
  
A non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171101.png" /> of the subsets of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171102.png" /> possessing the following properties: 1) its domain of definition is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171103.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171104.png" /> of Borel sets (cf. [[Borel set|Borel set]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171105.png" />, i.e. the smallest class of subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171106.png" /> containing the open sets and closed with respect to the set-theoretic operations performed a countable number of times; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171107.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171108.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b0171109.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711010.png" /> is countably additive. A Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711011.png" /> is called regular if
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{{MSC|28C15}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711012.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711013.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711014.png" /> of closed subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711015.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711016.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711017.png" /> with respect to which all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711018.png" /> are measurable. A Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711019.png" /> (or a Baire measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711020.png" />) is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711022.png" />-smooth if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711023.png" /> for any net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711024.png" /> of closed sets which satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711025.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711026.png" /> for any net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711027.png" /> of sets which are zero sets of continuous functions and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711028.png" />). A Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711029.png" /> (or Baire measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711030.png" />) is said to be tight if
+
[[Category:Set functions and measures on topological spaces]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711031.png" /></td> </tr></table>
+
===Definition===
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711032.png" /> is the class of compact subsets on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711033.png" /> (or
+
The terminology ''Borel measure'' is used by different authors with different meanings:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711034.png" /></td> </tr></table>
+
'''(A)''' Some authors use it for measures $\mu$ on the $\sigma$-algebra $\mathcal{B}$ of [[Borel set|Borel subsets]] of a given topological space $X$, i.e. functions $\mu:\mathcal{B}\to [0, \infty]$ which are countably additive.  
  
where
+
'''(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 {{Cite|Ha}} or Section 14 of Chapter 1 of {{Cite|Ro}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711035.png" /></td> </tr></table>
+
'''(C)''' Some authors (cp. with Definition 1.5(b) of {{Cite|Ma}} or with Section 1.1 of {{Cite|EG}}) use it for [[Outer measure|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).
  
Tightness and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711036.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711037.png" /> is a completely-regular Hausdorff space, then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711038.png" />-smooth (tight) finite Baire measure can be extended to a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711039.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711040.png" />-ring of sets generated by the compact (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017110/b01711041.png" />-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.
+
===Borel regular measures===
 +
In these three different contexts ''Borel regular measures'' are then defined as follows:
  
====References====
+
'''(A)''' Borel measures $\mu$ for which
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Varadarajan,  "Measures on topological spaces"  ''Transl. Amer. Math. Soc. Ser. 2'' , '''48'''  (1965) pp. 161–228  ''Mat. Sb.'' , '''55 (97)''' (1961) pp. 35–100</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Neveu,  "Bases mathématiques du calcul des probabilités" , Masson  (1970)</TD></TR></table>
+
\[
 +
\sup\; \{\mu (C):C\subset E\mbox{ is closed}\} = \mu (E) \qquad \mbox{for any Borel set } E.
 +
\]
  
 +
'''(B)''' Borel measures $\mu$ such that
 +
\[
 +
\sup\; \{\mu (C):C\subset E\mbox{  is compact}\} = \mu (E) \qquad \mbox{for any Borel set } E
 +
\]
 +
and
 +
\[
 +
\inf\; \{\mu (U):U\supset E\mbox{  is open}\} = \mu (E) \qquad \mbox{for any Borel set } E
 +
\]
 +
(cp. with Chapter 52 of {{Cite|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 {{Cite|Ma}}).
  
====Comments====
 
  
 +
''Warning'': The authors using terminology (A) call [[Tight measure|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 (directed set)|net]]) of closed sets with $F_k\downarrow \emptyset$.
  
====References====
+
''Warning'': The authors using terminology (C) use the term [[Radon measure|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 {{Cite|Ma}}).
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.L. Royden,   "Real analysis" , Macmillan  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.C. Zaanen,   "Integration" , North-Holland  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill (1953)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"W. Rudin,   "Real and complex analysis" , McGraw-Hill (1966) pp. 98</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.E. Taylor,   "General theory of functions and integration" , Blaisdell (1965)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.D. Aliprantz,   O. Burleinshaw,  "Principles of real analysis" , North-Holland (1981)</TD></TR></table>
+
 
 +
===Comments===
 +
The study of Borel measures is often connected with that of [[Baire measure|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 {{Cite|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 {{Cite|Ha}}).
 +
 
 +
If $X$ is a [[Completely-regular space|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 {{Cite|Ha}}).
 +
 
 +
===References===
 +
{|
 +
|-
 +
|valign="top"|{{Ref|AB}}|| C.D. Aliprantis, O. Burkinshaw,  "Principles of real analysis" , North-Holland (1981) {{MR|0607327}}  {{ZBL|0475.28001}}
 +
|-
 +
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}||    P. Mattila, "Geometry of sets  and measures in euclidean spaces".    Cambridge Studies in Advanced  Mathematics, 44. Cambridge University    Press, Cambridge,  1995.  {{MR|1333890}} {{ZBL|0911.28005}}
 +
|-
 +
|valign="top"|{{Ref|N}}|| J. Neveu, "Bases  mathématiques du calcul des probabilités" , Masson (1970) {{MR|0272004}} {{ZBL|0203.49901}}
 +
|-
 +
|valign="top"|{{Ref|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 {{MR|0148838}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Ro}}|| H.L. Royden, [[Royden, "Real analysis"|"Real analysis"]], Macmillan (1968) {{MR|0151555}} {{ZBL|0197.03501}}
 +
|-
 +
|valign="top"|{{Ref|R}}|| W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) {{MR|0055409}} {{ZBL|0052.05301}}
 +
|-
 +
|valign="top"|{{Ref|R2}}|| W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 {{MR|0210528}} {{ZBL|0142.01701}}
 +
|-
 +
|valign="top"|{{Ref|T}}|| A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) {{MR|0178100}} {{ZBL|0135.11301}}
 +
|-
 +
|valign="top"|{{Ref|Z}}|| A.C. Zaanen, "Integration" , North-Holland (1967) {{MR|0222234}} {{ZBL|0175.05002}}
 +
|-
 +
|}

Latest revision as of 09:39, 16 August 2013

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 \[ \sup\; \{\mu (C):C\subset E\mbox{ is closed}\} = \mu (E) \qquad \mbox{for any Borel set } E. \]

(B) Borel measures $\mu$ such that \[ \sup\; \{\mu (C):C\subset E\mbox{ is compact}\} = \mu (E) \qquad \mbox{for any Borel set } E \] and \[ \inf\; \{\mu (U):U\supset E\mbox{ is open}\} = \mu (E) \qquad \mbox{for any Borel set } E \] (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

[AB] C.D. Aliprantis, O. Burkinshaw, "Principles of real analysis" , North-Holland (1981) MR0607327 Zbl 0475.28001
[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
[R2] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 MR0210528 Zbl 0142.01701
[T] A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) MR0178100 Zbl 0135.11301
[Z] A.C. Zaanen, "Integration" , North-Holland (1967) MR0222234 Zbl 0175.05002
How to Cite This Entry:
Borel measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_measure&oldid=12636
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article