Products of two Borel measures
Author:
Roy A. Johnson
Journal:
Trans. Amer. Math. Soc. 269 (1982), 611625
MSC:
Primary 28C15; Secondary 03E35, 28A35
MathSciNet review:
637713
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Abstract: Let and be finite Borel measures on Hausdorff spaces and , respectively, and suppose product measures and are defined on the Borel sets of by integrating vertical and horizontal crosssection measure, respectively. Sufficient conditions are given so that and so that the usual product measure can be extended to a Borel measure on by means of completion. Examples are given to illustrate these ideas.
 [1]
Sterling
K. Berberian, Measure and integration, The Macmillan Co., New
York, 1965. MR
0183839 (32 #1315)
 [2]
S.
P. Franklin, Spaces in which sequences suffice, Fund. Math.
57 (1965), 107–115. MR 0180954
(31 #5184)
 [3]
D.
H. Fremlin, Products of Radon measures: a counterexample,
Canad. Math. Bull. 19 (1976), no. 3, 285–289.
MR
0435332 (55 #8292)
 [4]
R.
J. Gardner, The regularity of Borel measures and Borel
measurecompactness, Proc. London Math. Soc. (3) 30
(1975), 95–113. MR 0367145
(51 #3387)
 [5]
S.
L. Gulden, W.
M. Fleischman, and J.
H. Weston, Linearly ordered topological
spaces, Proc. Amer. Math. Soc. 24 (1970), 197–203. MR 0250272
(40 #3511), http://dx.doi.org/10.1090/S00029939197002502722
 [6]
Paul
R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New
York, N. Y., 1950. MR 0033869
(11,504d)
 [7]
Roy
A. Johnson, On product measures and Fubini’s
theorem in locally compact space, Trans. Amer.
Math. Soc. 123
(1966), 112–129. MR 0197669
(33 #5832), http://dx.doi.org/10.1090/S00029947196601976690
 [8]
Roy
A. Johnson, Measurability of cross section measure of a product
Borel set, J. Austral. Math. Soc. Ser. A 28 (1979),
no. 3, 346–352. MR 557285
(82k:28005)
 [9]
Roy
A. Johnson, Nearly Borel sets and product measures, Pacific J.
Math. 87 (1980), no. 1, 97–109. MR 590870
(82b:28024)
 [10]
I.
Juhász, Cardinal functions in topology, Mathematisch
Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S.
Kroonenberg; Mathematical Centre Tracts, No. 34. MR 0340021
(49 #4778)
 [11]
I.
Juhász, K.
Kunen, and M.
E. Rudin, Two more hereditarily separable nonLindelöf
spaces, Canad. J. Math. 28 (1976), no. 5,
998–1005. MR 0428245
(55 #1270)
 [12]
James
Keesling, Normality and properties related to
compactness in hyperspaces, Proc. Amer. Math.
Soc. 24 (1970),
760–766. MR 0253292
(40 #6507), http://dx.doi.org/10.1090/S00029939197002532927
 [13]
John
L. Kelley, General topology, D. Van Nostrand Company, Inc.,
TorontoNew YorkLondon, 1955. MR 0070144
(16,1136c)
 [14]
K. Kunen, A compact space under CH, Technical Report #21, University of Texas at Austin, 1980.
 [15]
E.
Marczewski and R.
Sikorski, Measures in nonseparable metric spaces, Colloquium
Math. 1 (1948), 133–139. MR 0025548
(10,23f)
 [16]
D.
A. Martin and R.
M. Solovay, Internal Cohen extensions, Ann. Math. Logic
2 (1970), no. 2, 143–178. MR 0270904
(42 #5787)
 [17]
A.
J. Ostaszewski, On countably compact, perfectly normal spaces,
J. London Math. Soc. (2) 14 (1976), no. 3,
505–516. MR 0438292
(55 #11210)
 [1]
 S. K. Berberian, Measure and integration, Macmillan, New York, 1965. MR 0183839 (32:1315)
 [2]
 S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107115. MR 0180954 (31:5184)
 [3]
 D. H. Fremlin, Products of Radon measures: a counterexample, Canad. Math. Bull. 19 (1976), 285289. MR 0435332 (55:8292)
 [4]
 R. J. Gardner, The regularity of Borel measures and Borel measurecompactness, Proc. London Math. Soc. (3) 30 (1975), 95113. MR 0367145 (51:3387)
 [5]
 S. L. Gulden, W. M. Fleischman and J. H. Weston, Linearly ordered topological spaces, Proc. Amer. Math. Soc. 24 (1970), 197203. MR 0250272 (40:3511)
 [6]
 P. R. Halmos, Measure theory, Van Nostrand, New York, 1950. MR 0033869 (11:504d)
 [7]
 R. A. Johnson, On product measures and Fubini's theorem in locally compact spaces, Trans. Amer. Math. Soc. 123 (1966), 112129. MR 0197669 (33:5832)
 [8]
 , Measurability of cross section measure of a product Borel set, J. Austral. Math. Soc. Ser. A 28 (1979), 346352. MR 557285 (82k:28005)
 [9]
 , Nearly Borel sets and product measures, Pacific J. Math. 87 (1980), 97109. MR 590870 (82b:28024)
 [10]
 I. Juhász, Cardinal functions in topology, Math. Centre Tracts, no. 34, Mathematisch Centrum, Amsterdam, 1971. MR 0340021 (49:4778)
 [11]
 I. Juhász, K. Kunen and M. E. Rudin, Two more hereditarily separable nonLindelöf spaces, Canad. J. Math. 28 (1976), 9981005. MR 0428245 (55:1270)
 [12]
 J. Keesling, Normality and properties related to compactness in hyperspaces, Proc. Amer. Math. Soc. 24 (1970), 760766. MR 0253292 (40:6507)
 [13]
 J. L. Kelley, General topology, Van Nostrand, New York, 1955. MR 0070144 (16:1136c)
 [14]
 K. Kunen, A compact space under CH, Technical Report #21, University of Texas at Austin, 1980.
 [15]
 E. Marczewski and R. Sikorski, Measures on nonseparable metric spaces, Colloq. Math. 1 (1948), 133139. MR 0025548 (10:23f)
 [16]
 D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143178. MR 0270904 (42:5787)
 [17]
 A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. 14 (1976), 505516. MR 0438292 (55:11210)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206377136
PII:
S 00029947(1982)06377136
Keywords:
Borel measure,
purely atomic measure,
separableregular Borel measure,
completion of a measure
Article copyright:
© Copyright 1982 American Mathematical Society
