Decomposability of Radon measures
Authors:
R. J. Gardner and W. F. Pfeffer
Journal:
Trans. Amer. Math. Soc. 283 (1984), 283293
MSC:
Primary 28C15
MathSciNet review:
735422
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Abstract: A topological space is called metacompact or metalindelöf if each open cover has a pointfinite or pointcountable refinement, respectively. It is well known that each Radon measure is expressible as a sum of Radon measures supported on a disjoint family of compact sets, called a concassage. If the unions of measurable subsets of the members of a concassage are also measurable, the Radon measure is called decomposable. We show that Radon measures in a metacompact space are always saturated, and therefore decomposable whenever they are complete. The previous statement is undecidable in ZFC if "metacompact" is replaced by "metalindelöf". The proofs are based on structure theorems for a concassage of a Radon measure. These theorems also show that the union of a concassage of a Radon measure in a metacompact space is a Borel set, which is paracompact in the subspace topology whenever the main space is regular.
 [F]
D.
H. Fremlin, Topological Riesz spaces and measure theory,
Cambridge University Press, London, 1974. MR 0454575
(56 #12824)
 [F]
D.
H. Fremlin, Topological measure spaces: two counterexamples,
Math. Proc. Cambridge Philos. Soc. 78 (1975),
95–106. MR
0377002 (51 #13177)
 [F]
D.
H. Fremlin, Decomposable measure spaces, Z. Wahrsch. Verw.
Gebiete 45 (1978), no. 2, 159–167. MR 510532
(80b:28003), http://dx.doi.org/10.1007/BF00715189
 [GP]
R.
J. Gardner and W.
F. Pfeffer, Are diffused, regular, Radon measures
𝜎finite?, J. London Math. Soc. (2) 20
(1979), no. 3, 485–494. MR 561140
(82i:28016), http://dx.doi.org/10.1112/jlms/s220.3.485
 [GP]
R.
J. Gardner and W.
F. Pfeffer, Some undecidability results concerning
Radon measures, Trans. Amer. Math. Soc.
259 (1980), no. 1,
65–74. MR
561823 (81e:54022), http://dx.doi.org/10.1090/S00029947198005618233
 [GP]
Kenneth
Kunen and Jerry
E. Vaughan (eds.), Handbook of settheoretic topology,
NorthHolland Publishing Co., Amsterdam, 1984. MR 776619
(85k:54001)
 [GrP]
G.
Gruenhage and W.
F. Pfeffer, When inner regularity of Borel measures implies
regularity, J. London Math. Soc. (2) 17 (1978),
no. 1, 165–171. MR 485446
(80c:28010), http://dx.doi.org/10.1112/jlms/s217.1.165
 [II]
A.
Ionescu Tulcea and C.
Ionescu Tulcea, Topics in the theory of lifting, Ergebnisse
der Mathematik und ihrer Grenzgebiete, Band 48, SpringerVerlag New York
Inc., New York, 1969. MR 0276438
(43 #2185)
 [J]
Roy
A. Johnson, Products of two Borel
measures, Trans. Amer. Math. Soc.
269 (1982), no. 2,
611–625. MR
637713 (82m:28026), http://dx.doi.org/10.1090/S00029947198206377136
 [JKR]
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)
 [Kn]
Kenneth
Kunen, A compact 𝐿space under CH, Topology Appl.
12 (1981), no. 3, 283–287. MR 623736
(82h:54065), http://dx.doi.org/10.1016/01668641(81)900067
 [Kr]
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
York, 1966. MR
0217751 (36 #840)
 [Ok]
Susumu
Okada, Supports of Borel measures, J. Austral. Math. Soc. Ser.
A 27 (1979), no. 2, 221–231. MR 531117
(80f:28018)
 [Os]
A.
J. Ostaszewski, On countably compact, perfectly normal spaces,
J. London Math. Soc. (2) 14 (1976), no. 3,
505–516. MR 0438292
(55 #11210)
 [P]
Washek
F. Pfeffer, Integrals and measures, Marcel Dekker Inc., New
York, 1977. Monographs and Textbooks in Pure and Applied Mathematics, Vol.
42. MR
0460580 (57 #573)
 [T]
Franklin
D. Tall, The countable chain condition versus
separability—applications of Martin’s axiom, General
Topology and Appl. 4 (1974), 315–339. MR 0423284
(54 #11264)
 [F]
 D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge Univ. Press, London, 1974. MR 0454575 (56:12824)
 [F]
 , Topological measure spaces: two counterexamples, Math. Proc. Cambridge Philos. Soc. 78 (1975), 95106. MR 0377002 (51:13177)
 [F]
 , Decomposable measure spaces, Z. Wahrsch. Verw. Gebiete 45 (1978), 159167. MR 510532 (80b:28003)
 [GP]
 R. J. Gardner and W. F. Pfeffer, Are diffused, regular, Radon measures finite?, J. London Math. Soc. (2) 20 (1979), 485494. MR 561140 (82i:28016)
 [GP]
 , Some undecidability results concerning Radon measures, Trans. Amer. Math. Soc. 259 (1980), 6574. MR 561823 (81e:54022)
 [GP]
 , Borel measures, Handbook of SetTheoretic Topology, NorthHolland, Amsterdam (to appear). MR 776619 (85k:54001)
 [GrP]
 G. Gruenhage and W. F. Pfeffer, When inner regularity of Borel measures implies regularity, J. London Math. Soc. (2) 17 (1978), 165171. MR 485446 (80c:28010)
 [II]
 A. Ionescu Tulcea and C. Ionescu Tulcea, Topics in the theory of lifting, SpringerVerlag, Berlin, Heidelberg and New York, 1969. MR 0276438 (43:2185)
 [J]
 R. A. Johnson, Products of two Borel measures, Trans. Amer. Math. Soc. 269 (1982), 611625. MR 637713 (82m:28026)
 [JKR]
 I. Juhász, K. Kunen and M. E. Rudin, Two more hereditarily separable nonLindelöf spaces, Canad. J. Math. 28 (1976), 9981005. MR 0428245 (55:1270)
 [Kn]
 K. Kunen, A compact space, Topology Appl. 12 (1981), 283287. MR 623736 (82h:54065)
 [Kr]
 K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966. MR 0217751 (36:840)
 [Ok]
 S. Okada, Supports of Borel measures, J. Austral. Math. Soc. Ser. A 27 (1979), 221231. MR 531117 (80f:28018)
 [Os]
 A. J. Ostaszewski, On countably compact, perfectly normal spaces, J. London Math. Soc. (2) 14 (1976), 505516. MR 0438292 (55:11210)
 [P]
 W. F. Pfeffer, Integrals and measures. Marcel Dekker, New York, 1977. MR 0460580 (57:573)
 [T]
 F. D. Tall, The countable chain condition versus separabilityapplications of Martin's axiom, Gen. Topology Appl. 4 (1974), 315339. MR 0423284 (54:11264)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407354228
PII:
S 00029947(1984)07354228
Keywords:
Radon measures,
Maharam measures,
decomposable measures,
metacompact and metalindelöf spaces,
weakly refinable spaces,
continuum hypothesis,
Martin's axiom
Article copyright:
© Copyright 1984 American Mathematical Society
