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Some Properties of Measure and Category

1981; American Mathematical Society; Volume: 266; Issue: 1 Linguagem: Inglês

10.2307/1998389

1088-6850

Arnold W. Miller,

Computability, Logic, AI Algorithms

Several elementary cardinal properties of measure and category on the real line are studied.For example, one property is that every set of real numbers of cardinality less than the continuum has measure zero.All of the properties are true if the continuum hypothesis is assumed.Several of the properties are shown to be connected with the properties of the set of functions from integers to integers partially ordered by eventual dominance.Several, but not all, combinations of these properties are shown to be consistent with the usual axioms of set theory.The main technique used is iterated forcing.Six properties of measure and category on the real line are studied.A(c) is the proposition that the union of fewer than continuum many meager sets is meager.B(c) says that the real line is not the union of fewer than continuum many meager sets.U(c) is the proposition that every subset of the real line of cardinality less than continuum is meager.A(m), B(m), and U(w) are defined analogously by replacing meager by measure zero.In the first section some equivalent forms of these properties are given, for example, it is shown that A(c) iff B(c) and every family of elements of to" of cardinality less than the continuum is eventually dominated by an element of ww.Characterizations of U(c) and B(c) are also given.In the second section we prove some theorems about unions of closed sets of measure zero, small dominating families, and strong measure zero sets.In the remaining sections several combinations of these properties are shown to be consistent with ZFC.These consistency results are summarized in the third section.The last section contains some open problems.I would like to thank K. Kunen for several helpful discussions.1.The properties and some of their equivalent forms.All the properties we consider are equivalent whether stated for 2", w", or the real line.For definiteness they will be stated for the Cantor space 2", so we will begin by reviewing the usual product topology and measure on 2" and also establish some standard terminology.For sets X and Y let Yx denote the set of functions from X into Y and \X\ denote cardinality of X.Let 2<u = U {2n : n < to} and similarly w<".Note that for s G 2<w, \s\ is the length of s when thought of as a sequence of zeros and ones.For s G 2<u let [s] = {x G 2" : s Ç. x (x extends s)}, then the usual product topology on 2" is given by taking {[s] : s G 2<w) as a basis for the open sets and the usual product measure ¡u. is given by letting /x([s]) = 2"|j|.For i and / finite sequences let -