On sigma-ideals of sets

Abstract

Let 4(DI') denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). THEOREM 1. (i) 4D and 'I' are a-ideals. (ii) The union of 41 or iP is I X I. (iii) The complement of each member of 4D or I contains a set of power c belonging to 4D and I, respectively. (iv) The unit square may be represented as the union of two complementary Borel sets: one in 4 and ' and the other one of Lebesgue measure zero and first category. (v) The unit square may be represented as the union of two complementary Borel sets: one in 4) and the other one in *I. THEOREM 2. 0(') does not satisfy (vi) There is a subclass T of power < c of the class 4(DI') such that every member of the class is contained in some member of the subclass. THEOREM 3. There does not exist a one-to-one mapping f from I X I onto itself, such that K E 4(DI) iffjK) is a Lebesgue measure zero (first category) subset of I X I. Theorems 2 and 3 hold for more general 4)(DI). A theorem on the theory of quotient (Boolean) algebras follows from these results. 1. On the Sierpin'ski-Erdos duality theorem. Let D denote the family of subsets of the unit square defined to be of first category in almost every vertical line in the sense of measure. To say in almost every vertical line in the sense of measure is equivalent to the statement: every vertical line except for a set whose corresponding projections to the x-axis is of measure zero in the unit interval. Let I denote the family of subsets of the unit square defined to be of measure zero in almost every vertical line in the sense of category. To say in almost every vertical line in the sense of category is equivalent to the statement: every vertical line except for a set whose corresponding projections to the x-axis is of first category in the unit interval. That is, 1.1 DEFINITION. K E P(J) if there is a measure zero (first category) set S in I (where I denotes the unit interval) such that KX is of first category (measure zero for every x in I S, (where KX = { Y E I: (x,y) E K, for some fixed x in I)). Without gain of generality the roles of vertical line and x-axis could have been interchanged with those of horizontal line and y-axis, respectively, Presented to the Society, January 21, 1976 in part under the title On the Sierpiniski-Erdos duality theorem; received by the editors March 25, 1976. AMS (MOS) subject classifications (1970). Primary 28A05, 28A10, 54A05, 04A05, 04AI5, 54C50, 54H05, 54H99; Secondary 02J04.