Powers of Transitive Bases of Measure and Category
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Abstract
We prove that on the real line the minimal cardinality of a base of measure zero sets equals the minimal cardinality of their transitive base.Next we show that it is relatively consistent that the minimal cardinality of a base of meager sets is greater than the minimal cardinality of their transitive base.We also prove that it is relatively consistent that the transitive additivity of measure zero sets is greater than the ordinary additivity and that the same is true about meager sets.0. Introduction.Let P be any group and J any ideal of subsets of P. We call a family s/ £ Ja base of ./iffor each / g J there is an A g j^such that I Q A. The minimal cardinality of a base of ./isdenoted by A(./).A family j/ ç ./iscalled a transitive base of J if for each / g J there exist A ^s# and p g P such that p ■ I ç A. The minimal cardinality of a transitive base of J is denoted by A,(./).The additivity of J', denoted by add(./ ), is the minimal cardinality of a family si Q ./suchthat Ují/ÍÍ.The transitive additivity, denoted by add,(J^), is the minimal cardinality of a set X ç P such that X • I <£ ./forsome / g J. Our set theory is ZFC.We use the standard set theoretical notation.In particular, u is the set of all natural numbers, ^(w) is the set of all subsets of « and "u is the set of all functions from u to w.An ordinal is the set of proceeding ordinals, e.g. if n is a natural number, then n = {0,\,...,n -1}.If A' is any set and k any cardinal, then let \X\ denote the cardinality of X, [X]K the set of all subsets of X of cardinality k and [X]*K the set of all subsets of X of cardinality not greater than k.Let R denote the real line with its usual topology and algebraic structure.Let Q stand for rationals and Z for integers.We fix some enumeration {qn: n g co} of Q.Let p denote the Lebesque measure on R. If p, r g R and p < r, then let [p; r] denote the closed interval and (p; r) the open interval with endpointsp, r.Let 3nx mean "there is infinitely many natural «" and V«00 mean "for all but finitely many natural «." For/, g e "« we write/ < g if for any n < w, we have/(«) < g(n) and we write / -< g if V«00 /(«) < g(n).A subset D ç uu is a dominating family if for any function / g uu there is d G D with / < d.The minimal cardinality of a dominating family is denoted by A. A subset A ç "u is unbounded if there is no / g "w such that for any g g A we have g < f.The minimal cardinality of an unbounded family is denoted by \.