The Erd¨os-Sierpi´nskiDuality Theorem Shingo SAITO¤ 0 Notation The Lebesgue measure on R is denoted by ¹. The σ-ideals that consist of all meagre subsets and null subsets of R are denoted by M and N respectively. 1 Similarities between Meagre Sets and Null Sets Definition 1.1 Let I be an ideal on a set. A subset B of I is called a base for I if each set in I is contained in some set in B. Proposition 1.2 Each meagre subset of a topological space is contained in some meagre Fσ set. In particular, M\Fσ is a base for M. S1 Proof. Let A be a meagre subset of a topological space. Then A = n=1 An for some S1 nowhere dense sets An. The set n=1 An, which contains A, is meagre and Fσ since the sets An are nowhere dense and closed. Proposition 1.3 Each null subset of R is contained in some null G± set. In other words, N\G± is a base for N . Proof. This is immediate from the regularity of the Lebesgue measure. Proposition 1.4 Every uncountable G± subset of R contains a nowhere dense null closed set with cardinality 2!. T1 Proof. Let G be an uncountable G± set. Then G = n=0 Un for some open sets Un. <! We may construct a Cantor scheme f Is j s 2 2 g such that Is is a compact nonde- ¡n <! ! generate interval contained in Ujsj with ¹(Is) 5 3 for every s 2 2 . Let f : 2 ¡! R © ª T1 denote the associated map of the Cantor scheme defined by f(®) = n=1 I®jn . Denote the range of f by A. T1 S Note that A = n=1 s22n Is, which implies that A is closed. Moreover A is null because à ! [ X 2n ¹(A) 5 ¹ I 5 ¹(I ) 5 ! 0 as n ! 1: s s 3n s22n s22n ¤[email protected] 1 Thus Int A = ;, which shows that A is nowhere dense since A is closed. The injectivity ! <! of f shows that jAj = 2 . Since Is ½ Ujsj for every s 2 2 , we have \1 [ \1 A = Is ½ Un = G: n=1 s22n n=1 Corollary 1.5 Every residual subset of R contains a meagre set with cardinality 2!. c S1 Proof. Let A be a residual subset of R. Then A = n=1 An for some nowhere dense S1 c S1 c sets An. Since n=1 An is a meagre Fσ set that contains A , the set ( n=1 An) is a S1 c nonmeagre G± set contained in A. Therefore Proposition 1.4 shows that ( n=1 An) contains a meagre set with cardinality 2!, which is contained in A. Corollary 1.6 The complement of each null subset of R contains a null set with cardi- nality 2!. Proof. Let A be a subset of R with Ac 2 N . Then A is measurable and ¹(A) = 1. Therefore the regularity of ¹ yields a closed set F contained in A with ¹(F ) = 1. It follows from Proposition 1.4 that F contains a null set with cardinality 2!, which is contained in A. 2 Erd¨os-Sierpi´nskiDuality Theorem Proposition 2.1 There exist a meagre Fσ subset A and a null G± subset B of R that satisfy A \ B = ; and A [ B = R. S1 ¡n¡j Proof. Enumerate Q = fq1; q2;:::g, and put Pn = j=1 B(qj; 2 ) for positive inte- T1 c gers n. Define B = n=1 Pn and A = B . Since Pn is open for every positive integer n, we have B 2 G± and A 2 Fσ. We shall prove that A is meagre and B is null. The set B is null because X1 X1 ¡ ¡n¡j ¢ ¡n¡j+1 ¹(B) 5 ¹(Pn) 5 ¹ B(qj; 2 ) = 2 j=1 j=1 = 2¡n+1 ! 0 as n ! 1: c For every positive integer n, the set Pn is nowhere dense because Pn is open and S1 c dense. It follows from A = n=1 Pn that A is meagre. Theorem 2.2 (Erd¨os-Sierpi´nskiDuality Theorem) Assume that the continuum hy- pothesis holds. Then there exists an involution f : R ¡! R such that f(A) is meagre if and only if A is null, and f(A) is null if and only if A is meagre for every subset A of R. 2 ! ! Proof. Since jM \ Fσj = 2 and jN \ G±j = 2 , it follows from Proposition 2.1 that ! ! there exist bijections » 7¡! A» from 2 to M\Fσ, and » 7¡! B» from 2 to N\G± that satisfy A0 \ B0 = ; and A0 [ B0 = R. ! Define inductively a map » 7¡! F» from 2 to M such that (1) F0 = A0; c (2) F»+1 is the union of F» [A» and a meagre set contained in (F» [A») with cardinality 2! for every » 2 2!; S ! (3) F» = ®2» F® for every limit ordinal » 2 2 . We may construct such a map due to Corollary 1.5 and the continuum hypothesis. Similarly Corollary 1.6 and the continuum hypothesis allow us to define a map » 7¡! ! G» from 2 to N such that (1) G0 = B0; c (2) G»+1 is the union of G» [ B» and a null set contained in (G» [ B») with cardinality 2! for every » 2 2!; S ! (3) G» = ®2» G® for every limit ordinal » 2 2 . For each » 2 2!, there exists a bijection f : F nF ¡! G nG since jF nF j = S » »+1 » »+1 S» »+1 » jG n G j = 2!. Put fe = f . Then fe is a bijection from (F n F ) = »+1 » S »22! » »22! »+1 » e e¡1 R n F0 = G0 to »22! (G»+1 n G») = R n G0 = F0. Thus the union f of f and f is an involution from R to R. Let M be a meagre set. Proposition 1.2 implies that M is contained in A»0 for some ! »0 2 2 . Then [ M ½ A»0 ½ F»0+1 = (F»+1 n F») [ F0 »2»0+1 shows that à ! [ [ f(M) ½ f (F»+1 n F») [ F0 = f(F»+1 n F») [ f(F0) »2» +1 »2» +1 [ 0 0 = (G»+1 n G») [ G0 = G»0+1 2 N ; »2»0+1 which implies that f(M) is null. Similarly it follows from Proposition 1.3 that f(N) is meagre for every null set N. Since f is an involution, we conclude that f(M) is null only if M is meagre, and that f(N) is meagre only if N is null. Remark. Assuming the continuum hypothesis is too much; the proof of Theorem 2.2 works on the mere assumption that add(M) = add(N ) = 2!. In particular, assuming the Martin axiom is enough. 3 3 References [1] Tomek Bartoszy´nskiand Haim Judah, Set Theory—On the Structure of the Real Line, A K Peters. [2] Winfried Just and Martin Weese, Discovering Modern Set Theory II, Graduate Studies in Mathematics, 18, American Mathematical Society. [3] John C. Oxtoby, Measure and Category, Graduate Texts in Mathematics, 2, Springer- Verlag. [4] S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, 180, Spring-Verlag. 4.