Hereditarily additive families in descriptive set theory and Borel measurable multimaps
Author:
Roger W. Hansell
Journal:
Trans. Amer. Math. Soc. 278 (1983), 725749
MSC:
Primary 54H05; Secondary 04A15, 28A05
MathSciNet review:
701521
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A family of Borel subsets of a space is (boundedly) Borel additive if, for some countable ordinal , the union of every subfamily of is a Borel set of class in . A problem which arises frequently in nonseparable descriptive set theory is to find conditions under which this property is "hereditary" in the sense that any selection of a Borel subset from each member of (of uniform bounded class) will again be a Borel additive family. Similar problems arise for other classes of projective sets; in particular, for Souslin sets and their complements. Positive solutions to the problem have previously been obtained by the author and others when is a complete metric space or under additional settheoretic axioms. We give here a fairly general solution to the problem, without any additional axioms or completeness assumptions, for an abstract "descriptive class" in the setting of generalized metric spaces (e.g., spaces with a pointfinite open base). A typical corollary states that any pointfinite (co) Souslin additive family in (say) a metrizable space is hereditarily (co) Souslin additive. (There exists a pointcountable additive family of subsets of the real line which has a point selection which is not even Souslin additive.) Two structure theorems for "hereditarily additive" families are proven, and these are used to obtain a nonseparable extension of the fundamental measurable selection theorem of Kuratowski and RyllNardzewski, and a complete solution to the problem of Kuratowski on the Borel measurability of complex and product mappings for nonseparable metric spaces.
 [1]
William
G. Fleissner, An axiom for nonseparable Borel
theory, Trans. Amer. Math. Soc. 251 (1979), 309–328. MR 531982
(83c:03043), http://dx.doi.org/10.1090/S00029947197905319829
 [2]
, Square of sets, preprint.
 [3]
William
G. Fleissner, Roger
W. Hansell, and Heikki
J. K. Junnila, PMEA implies proposition 𝑃, Topology
Appl. 13 (1982), no. 3, 255–262. MR 651508
(83f:54042), http://dx.doi.org/10.1016/01668641(82)900347
 [4]
D.
H. Fremlin, R.
W. Hansell, and H.
J. K. Junnila, Borel functions of bounded
class, Trans. Amer. Math. Soc.
277 (1983), no. 2,
835–849. MR
694392 (84d:54067), http://dx.doi.org/10.1090/S00029947198306943920
 [5]
Roger
W. Hansell, On the nonseparable theory of 𝑘Borel and
𝑘Souslin sets, General Topology and Appl. 3
(1973), 161–195. MR 0319170
(47 #7716)
 [6]
R.
W. Hansell, On characterizing nonseparable analytic and extended
Borel sets as types of continuous images, Proc. London Math. Soc. (3)
28 (1974), 683–699. MR 0362269
(50 #14711)
 [7]
R.
W. Hansell, On the representation of nonseparable
analytic sets, Proc. Amer. Math. Soc. 39 (1973), 402–408.
MR
0380752 (52 #1649), http://dx.doi.org/10.1090/S00029939197303807523
 [8]
R.
W. Hansell, Some consequences of
(𝑉=𝐿) in the theory of analytic sets, Proc. Amer. Math. Soc. 80 (1980), no. 2, 311–319. MR 577766
(81j:54058), http://dx.doi.org/10.1090/S00029939198005777660
 [9]
R.
W. Hansell, Boreladditive families and Borel maps in metric
spaces, General topology and modern analysis (Proc. Conf., Univ.
California, Riverside, Calif., 1980) Academic Press, New YorkLondon,
1981, pp. 405–416. MR 619067
(82i:54069)
 [10]
R.
W. Hansell, Borel measurable mappings for
nonseparable metric spaces, Trans. Amer. Math.
Soc. 161 (1971),
145–169. MR 0288228
(44 #5426), http://dx.doi.org/10.1090/S00029947197102882281
 [11]
J.
Kaniewski and R.
Pol, Borelmeasurable selectors for compactvalued mappings in the
nonseparable case, Bull. Acad. Polon. Sci. Sér. Sci. Math.
Astronom. Phys. 23 (1975), no. 10, 1043–1050
(English, with Russian summary). MR 0410657
(53 #14405)
 [12]
K. Kuratowski, Quelques problèmes concernant espaces métriques nonséparables, Fund. Math. 25 (1935), 532545.
 [13]
Gustave
Choquet, Topology, Translated from the French by Amiel
Feinstein. Pure and Applied Mathematics, Vol. XIX, Academic Press, New
YorkLondon, 1966. MR 0193605
(33 #1823)
 [14]
K.
Kuratowski and C.
RyllNardzewski, A general theorem on selectors, Bull. Acad.
Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13
(1965), 397–403 (English, with Russian summary). MR 0188994
(32 #6421)
 [15]
Kazimierz
Kuratowski and Andrzej
Mostowski, Set theory, Second, completely revised edition,
NorthHolland Publishing Co., AmsterdamNew YorkOxford; PWN—Polish
Scientific Publishers, Warsaw, 1976. With an introduction to descriptive
set theory; Translated from the 1966 Polish original; Studies in Logic and
the Foundations of Mathematics, Vol. 86. MR 0485384
(58 #5230)
 [16]
S.
J. Leese, Measurable selections and the uniformization of Souslin
sets, Amer. J. Math. 100 (1978), no. 1,
19–41. MR
0507445 (58 #22447)
 [17]
E.
Michael, On maps related to 𝜎locally finite and
𝜎discrete collections of sets, Pacific J. Math.
98 (1982), no. 1, 139–152. MR 644945
(83b:54012)
 [18]
A.
H. Stone, Nonseparable Borel sets, Rozprawy Mat.
28 (1962), 41. MR 0152457
(27 #2435)
 [19]
A.
H. Stone, Some problems of measurability, Topology Conference
(Virginia Polytech. Inst. and State Univ., Blackburg, Va., 1973)
Springer, Berlin, 1974, pp. 242–248. Lecture Notes in Math.,
Vol. 375. MR
0364589 (51 #843)
 [1]
 W. G. Fleissner, An axiom for nonseparable Borel theory, Trans. Amer. Math. Soc. 251 (1979), 309328. MR 531982 (83c:03043)
 [2]
 , Square of sets, preprint.
 [3]
 W. G. Fleissner, R. W. Hansell and H. J. K. Junnila, PMEA implies position , Topology Appl. 13 (1982), 255262. MR 651508 (83f:54042)
 [4]
 D. H. Fremlin, R. W. Hansell and H. J. K. Junnila, Borel functions of bounded class, Trans. Amer. Math. Soc. (to appear). MR 694392 (84d:54067)
 [5]
 R. W. Hansell, On the nonseparable theory of Borel and Souslin sets, General Topology Appl. 3 (1973), 161195. MR 0319170 (47:7716)
 [6]
 , On characterizing nonseparable analytic and extended Borel sets as types of continuous images, Proc. London Math. Soc. (3) 28 (1974), 683699. MR 0362269 (50:14711)
 [7]
 , On the representation of nonseparable analytic sets, Proc. Amer. Math. Soc. 39 (1973), 402408. MR 0380752 (52:1649)
 [8]
 , Some consequences of in the theory of analytic sets, Proc. Amer. Math. Soc. 80 (1980), 311319. MR 577766 (81j:54058)
 [9]
 , Boreladditive families and Borel maps in metric spaces, General Topology and Modern Analysis (L. F. McAuley and M. M. Rao, eds.), Academic Press, New York, 1981. MR 619067 (82i:54069)
 [10]
 , Borel measurable mappings for nonseparable metric spaces, Trans. Amer. Math. Soc. 161 (1971), 145169. MR 0288228 (44:5426)
 [11]
 J. Kaniewski and R. Pol, Borel measurable selections for compactvalued mappings in the nonseparable case, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 19431050. MR 0410657 (53:14405)
 [12]
 K. Kuratowski, Quelques problèmes concernant espaces métriques nonséparables, Fund. Math. 25 (1935), 532545.
 [13]
 , Topology, Vol. 1, Academic Press, New York, PWN, Warsaw, 1966. MR 0193605 (33:1823)
 [14]
 K. Kuratowski and C. RyllNardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397403. MR 0188994 (32:6421)
 [15]
 K. Kuratowski and A. Mostowski, Set theory (with an introduction to descriptive set theory), NorthHolland, Amsterdam, 1976. MR 0485384 (58:5230)
 [16]
 S. J. Leese, Measurable selections and uniformization of Suslin sets, Amer. J. Math. 100 (1978), 1941. MR 0507445 (58:22447)
 [17]
 E. Michael, On maps related to locally finite and discrete collections of sets, Pacific J. Math. 98 (1982), 139152. MR 644945 (83b:54012)
 [18]
 A. H. Stone, Nonseparable Borel sets, Rozprawy Mat. 28 (1962). MR 0152457 (27:2435)
 [19]
 , Some problems of measurability (Topology Conf., Blacksburg, Va., 1973), Lecture Notes in Math., no. 375, SpringerVerlag, Berlin and New York, 1974, pp. 242248. MR 0364589 (51:843)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
54H05,
04A15,
28A05
Retrieve articles in all journals
with MSC:
54H05,
04A15,
28A05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198307015219
PII:
S 00029947(1983)07015219
Keywords:
Hereditarilyadditive families,
descriptive set theory,
Borel measurable multimaps,
measurable selectors,
generalized metric spaces,
pointfinite family
Article copyright:
© Copyright 1983
American Mathematical Society
