On some problem of A. Rosłanowski

Szymon Plewik
1996 Colloquium Mathematicum  
We present a negative answer to problem 3.7(b) posed on page 193 of [2] , where, in fact, A. Ros lanowski asked: Does every set of Lebesgue measure zero belong to some Mycielski ideal ? We identify a set X ∈ [ω] ω with its characteristic function, i.e. with the sequence (X(0), X(1), . . .) ∈ 2 ω such that X(n) = 1 iff n ∈ X. A set X ∈ [ω] ω has asymptotic density d whenever where |X ∩ n| denotes the number of natural numbers from X less than n. We consider the family of all sets of asymptotic
more » ... nsity not equal to 1/2, i.e. the set An old result of E. Borel [1] says: The set A has Lebesgue measure zero. A direct consequence of this result is Theorem. The set A does not belong to any Mycielski ideal. P r o o f. Our notation follows [2]. If K is a normal system, i.e. for each X ∈ K there exist two disjoint subsets of X which belong to K, then K contains three disjoint sets X, Y and Z. Since |X ∩ n| + |Y ∩ n| + |Z ∩ n| ≤ n, one of the sets: X, Y or Z does not contain any subset of asymptotic density 1/2. Suppose X is a such set. If Player I always chooses zero, then he wins the game Γ (X, A), because any set (sequence) which can be the result of that game is not of asymptotic density 1/2 and thus belongs to A. This means that the set A does not belong to the Mycielski ideal generated by K. If one considers Mycielski ideals on k ω , where k > 2 is a natural number, then our theorem can be slightly modified. The Lebesgue measure and 1991 Mathematics Subject Classification: 03E05, 04A20, 28A05. [297]
doi:10.4064/cm-69-2-297-298 fatcat:plwnsh2ybnamfosmcrye3neasq