Frolík, Z.

Holický, P.

Institute of Mathematics AS CR, v. v. i.

WSAA 8Zdeněk Frolík; Petr HolickýNon-separable analytic spaces and measurabilityIn: Zdeněk Frolík (ed.): Abstracta. 8th Winter School on Abstract Analysis.Czechoslovak Academy of Sciences, Praha, 1980. pp. 73--78.Persistent URL: http://dml.cz/dmlcz/701180Terms of use:© Institute of Mathematics of the Academy of Sciences of the Czech Republic,1980Institute of Mathematics of the Academy of Sciences of the Czech Republicprovides access to digitized documents strictly for personal use. Each copy of anypart of this document must contain these Terms of use.This paper has been digitized, optimized for electronic deliveryand stamped with digital signature within the project DML-CZ:The Czech Digital Mathematics Library http://project.dml.czEIGHTH WINTER SCHOOL ON ABSTRACT ANALYSIS (1980) ' 7 3 Non-separable analytic spaces and meaeurability Z. Frolik and P. Holick? In [lj we introduced the concepts of 0"-discrete decompo-sability (^-dd) and hypsranalyticity in uniform spaces. We stated Theorem 1 which clarifies the connection between meaeu­rability and discrstnes80 and Theorem 2 which extends the clas­sical (first) separation principle. These results which extend the original Hansell'e reeults for complete metric spaces were eupplemented and prepared for publication ([<-{]« [3] f [4]. [?])• We choosed an essential part of our results which concern completely regular topological epaces. Since the definitions and terminology are slightly modified, let us start with nota­tions and definitions. All spaces are assumed to be completely regular topological spaces. The letters X,Y stand always for such epaces, the letter # stands for any infinite cardinal through theee notes. Definitions (a) A family { X J B G A } of subsets of .'JC is called tf-dd x a - U { x 2 . | n G o ; }1 > i f there are x"CXQ s.t. a a аnd { x £ | a G A } ars discrete.2) (b) A (eet-valued) mapping f : X—*Y is called #-dd-preserv-ing i f { f [ x a ] } is tf-dd in Y whenever { x f l } is ^tf-dd *> to - { 0 , 1 , 2 , . . . } ' discrete • discrots in some continuous pseudometric through this note I 74 in X . (c) X is called ^-analytic if there is a compact-valued tf'-dd-preserving mapping f from a complete metric apace M of weight lees or equal £ onto X s.t. rl[c] • {n^N | 'Wnc j- 0} is closed in M whenever C is closed in X . Such triples (f,M,X) are called ^-analytic parametrizatlone of X , (d) The elements of the smallest <7-algebra closed under dis-crete unions of cardinality .dt and containing zero sate of con-tinuous real functions are called 46-Baire sets. Theorems 1 and 2 of [l] are still valid if we write %--analytic or £-Baire instead of hyperanalytlc or hyper-Baire, respectively* We are going to formulate several results concerning cha-racterizations of #-analytic spacee. Images of measurable sete, and meaeurable selections* Characterizations Theorem 1« The following etatemente concerning .X are equiva-lent: (a) X is «. -analytic. (b) X is a Suslln subset of K*M where K ie compact and M ia a complete metric space of weight fr • (c) X ie a paracoropact Suslin subset of a Cech complete space of weight & • til (d) There is a 2-analytic parametrlzatlon (f, at ,X) where & ie (a set of cardinality X, } endowed with discrete metric-The condition (d) hae an auxiliary character for obtaining other characterizations and results, and unables to extend the 75 claeeical methode which are baeed upon the connection between 0) -analytic epacee and irrational numbere ( (ou ) . Ueing eeeentially Theoreme 1 and 2 from [lj we derived re-eulte concerning Imagee of meaeurable eete Theorem 2. Let X be a %-analytic epacer P a metric epace, and f : X—>P a (f -dd-preeerving mapping e.t. f~ [s] ie Suelin in X whenever S ie Suelin in P . Then: (a) f"1^] ia #-Baire in X if B ie Borel (- Baire) in P. (b) The graph of f , i.e. the eet {(xiP)€X*P | PGf(x)} . ie X-analytic in X*P . (c) The eet f[x] ie ^-analytic in P . (d) If f"1^] ia a je-Baire eet in X then B ie te-Baire in f[x] . The etatement (d) eeye that f ia "%-Baire-quotient* (cf. [e]). The- following characterization of "bimeaaurable" mappinge between complete metric epacee and Theorem 2 glvee a good pictu-re of meaeurable Imagee of Borel («- Baire) eete in the caae of complete metric epacee. Theorem 3. Let M,P be complete metric epacee of weight < & and let f : M—*P be a to -Baire meaeurable 6 -dd-preeerving mapping. Then f ie #-Baire meaeurable, and f i8 at-Baire meaeurable iff the family { { p } | f (P) i s uncountable] i8 (f-dd (i.e. ite union ie * •rf-diecrete"). The eufficiency of the laet condition can be proved eimilar-ly ae the well-known analogy in eeparable epacee (N. Luzin) ueing Theoreme 1 and 2 from [l] , and the "eecond eeparation principle" : 76 ([7] . Th. 14). The other implication could be reduced to the eeparable ca-ee in the following way: We find an analogy to a Mazurkiewicz--Sierpineki theorem [8j v Th. 2, p. 506 , and prove that U • {p I f-1(p) ie uncountable } is St -analytic. If it were not 4-discrete we find an uncoun-table compact subset KCU . Uelng Theorem 2 from [lj again we ehow that f" [KJ is a separable Borel subset of H , and we can U8e Purves'e reeult ([9J 0 Theorem). An other object of our inveetigation are Selectiona We prove analogies to von Neumann ([10] , Lemma 16) and Mazurkiewicz ([8J , Th. 3, p. 491) theorems: Lemma. Let P be a cloeed eubspace of at (c.f. Theorem 1(d)), and let h be a contlnuoue 6 -dd-preeerving mapping of P on-to X • Consider the eelection 8 : X—»P s.t. a(x) ie the - 1 (O emalleat (in lexicographical order) element of h (x)C # • Then (a) 8-l[6]G(^\^H3) (b) P\s[x] is Suslin in P • The following eelection theorem can be easily derived: Theorem 4. Let F : X—*Y be a set-valued mapping, and let the graph of F be point-analytic. ' Let the projection of graph 3) y \ y etande for differences of Suslin subsets of X ; ( . /N i f 5 )tf-d denotes d -discrete unions of elements of 4^ There ie an analytic parametrizatlon (f,Mf graph F) where f ie a eingle-valued correepondence 77 F into X is <?-dd-preserving. Then F admits a selection s.t. F - 1 [ G ] E ( Y\*f )d-d for open G , and if graph F is point-Luzin ' then the selection f can be chosen s.t. more-over, graph FNgraph f is Suslln. Theorem 1 from [lj implies that compact-valued Suslin-mea-surabls mapping from an analytic space into a metric space ful-fills the assumptions of Theorem 4. The method of Kuratowski and Ryll-Nardzswskl [llj can be also used for derivation of eelection theorems. For an example 866 [l2]* ' There i s an ana ly t ic pararaetrization ( f ,M, graph F) where f i s a one-to-one mapping References [li Z. Frolik and P. Holický: On the non-separable descriptive theory. Fifth Winter School On Abstract Analysis, Abstrac ta, Praha 1977, p. 21-22 [2] Z. Frolik and P. Holický: Decomposability of completely Suslin-additivs families; submitted to Proc. AMS [3] Z. Frolik and P. Holický: Analytic and Luzln spaces (non--eeparable case); submitted to Topology and its appli-cations [4] Z. Frolik and P. Holický: Applications of Luzinian separa-tion principles (non-separable case); submitted to Fund. Math. [5] Z, Frolik and P. Holický: Selectione via ordering;in prepa-ration [e] z. Frolik: A measurable map with analytic domain and metrl-zable range is quotient. Bull.Araer.Math.Soc. 76, 1970, p. 1112-1117 [7] C.A. Rogers and R.C Willmott: On the uniformizatlon of sete in topological spaces. Acta Math, 120, 1968 [8J K. Kuratowski: Topologija 1. Mir, Moskva 1966 78 [9] R. Purvea: Blmeasurable functions* Fund.Nath. 58, 1966, p. 149-157 [lOj 3. von Neumann: On rings of operators. Red.Th» In: Collec-ted Works..Vol. Ill [ill K. Kuratowski and C. Ryll-Nardzewski: A general theorem on selectors. Bull.Acad.Polon.Sci., Ser.Math. Astronom. Phys. 13, 1965, p. 397-403 [l2j 3. Kaniewski and R. Pol: Borel-measurable selectors for compact-valued mappings in the non-separable case. Bull. Acad.Polon., Vol. XXIII,.No. 10, 1975 

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Non-separable analytic spaces and measurability

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Non-separable analytic spaces and measurability

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