Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 078.04203
Autor: Erdös, Pál; Fodor, G.
Title: Some remarks on set theory. IV. (In English)
Source: Acta Sci. Math. 18, 243-260 (1957).
Review: [Part V see Zbl 072.04103]
The paper is dealing with relations in a given set E of cardinality m > \aleph0. For every x in E let R(x) be a subset of E. Two distinct elements x,y of E are independent, if both x in R(y) und y in R(x). Every monopunctual subset of E as well as every subset of pairwise independent points is called a free subset of R. The following 3 conditions are considered:
(A) There exists a cardinality n < m satisfying kR(x) < n for every x in E (kX means the cardinality of X).
(B) E is a metric space and dist(x,R(x)) > 0 for every x in E.
(C) There exists a real number r > 0 such that, putting g(x) = dist(x,R(x)), the set {x | g(x) \geq r} contains in B a subset of positive measure; B denotes the system of all Borel sets of E; E is a metric space containing an everywhere dense set of a cardinality < i (i denotes the first inaccessible cardinal number > \aleph0). For a system S of sets a subsystem I of S is called a p-additive ideal provided (I) the union of every subsystem of I of cardinality < p belongs to I and (II) for every X in I the relations Y \subseteq X, Y in S imply Y in I.
Theorem 1. If m = \aleph\gamma > i and if I denotes a proper \aleph\lambda+1-ideal of subsets of E such that {x} in I for every x in E; if B \subseteq E, B\not in I, then there exists a disjointed \omega\lambda+1-sequence B\xi of subsets of E such that B\xi \not in I (\xi < \omega\lambda+1) and B = \bigcup B\xi.
This theorem is used in providing the following one (theorem 3): Under the conditions of Th. 1. if R(x) is finite for every x in E, then for every \omega-sequence of subsets E\xi of E such that E\xi \not in I, there exists a free subset E' of E satisfying E' \cap E\xi \not in I for every \xi < \omega. Now suppose that the condition (B) holds. Let E denote the set of all real numbers and kR(x) < \aleph0 (x in E). Then there exists a freesubset E' of E such that E' be everywhere of the second category and that the Lebesgue outer measure \mu(E') of E' be b-a in every interval (a,b) (Th. 6). Let not E be an interval of real numbers and B be a \sigma-algebra of subsets of E containing all subintervals of E; let \mu be a non-trivial measure on B. Then the condition (C) implies the existence in B of a free subset of positive \mu-measure.
In the particular case when R(x) is the complement of an interval of E whose center is at x, the converse holds too: the existence in B of a free subset of E of positive \mu-measure implies the condition (C) (Th. 7). Theorem 11: Let K be a disjointed class of cardinality g of subsets of E of cardinality m = kE each; then the condition (A) implies the existence of a free subset E' of E such that the cardinality of X \cap E' be m for every X in K (here m \geq \aleph0).
Reviewer: G.Kurepa
Index Words: Set Theory
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