Cardinal Functions of Purely Atomic Measures
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Results in Mathematics
Cardinal Functions of Purely Atomic Measures Szymon Gla¸b
and Jacek Marchwicki
Abstract. Let μ be a purely atomic finite measure. Without loss of generality we may assume that μ is defined on N, and the atoms with smaller indexes have larger masses, that is μ({k}) ≥ μ({k + 1}) for k ∈ N. By fµ : [0, ∞) → {0, 1, 2, . . . , ω, c} we denote its cardinal function fµ (t) = |{A ⊂ N : μ(A) = t}|. We study the problem for which sets R ⊂ {0, 1, 2, . . . , ω, c} there is a measure μ such that R = rng(fµ ). We are also interested in the set-theoretic and topological properties of the set of μ-values which are obtained uniquely. Mathematics Subject Classification. Primary: 40A05, Secondary: 11K31. Keywords. purely atomic measure, achievement set, set of subsums, absolutely convergent series.
1. Introduction Let μ be a purely atomic finite measure. Following [5] we may assume that μ is defined on N := {1, 2, 3, . . . } and μ({k}) ≥ μ({k + 1}) > 0. Throughout the paper we assume that measures are always of this type. By the range of μ we understand the set rng(μ) := {μ(E) : E ⊂ N}. To simplify the notation let xn = μ({n}) be a measure of the n-th largest atom of μ. Note that ∞ N . εn xn : (εn ) ∈ {0, 1} rng(μ) = n=1
The latter set is also denoted by A(xn ) and it is called the achievement set N of (xn ) (see ∞ [12]). We can identify a measure μ with the function {0, 1} (εn ) → n=1 εn xn . This is the continuous mapping from the Cantor space {0, 1}N to the real line, and therefore μ−1 (t) is a closed subset of the Cantor 0123456789().: V,-vol
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S. Gla ¸ b and J. Marchwicki
Results Math
space {0, 1}N . Thus the cardinality |μ−1 (t)| belongs to the set {0, 1, 2, . . . , ω, c} where ω stands for the first infinite cardinal while c for the continuum [14, Section 6]. By fμ we denote the cardinal function fμ (t) = |μ−1 (t)| = |{A ⊂ N : μ(A) ∞= t}|; we will write f instead of fμ if it is clear which measure μ or series n=1 xn is considered. By R(fμ ) (or R(f )) we denote the range of the cardinal function fμ , that is the set of all cardinals |μ−1 (t)| where t ∈ rng(μ). Hence R(f ) ⊂ {1, 2, . . . , ω, c}. For example, n ∈ R(fμ ) means that there are n distinct sets A1 , . . . , An ⊂ N such that μ(Ai ) = μ(A1 ) for i = 2, . . . , n and μ(A) = μ(A1 ) for any A ∈ P(N)\{A1 , . . . , An }, where P(N) stands for the family of all subsets of N. If fμ (t) = 1, then we say that t is uniquely obtained, (or it has unique representation or unique expansion). The set fμ−1 (1) will be called a set of uniqueness for μ. By F in we denote the collection of all finite subsets of N. Identifying subset of N with its characteristic functions, we consider on P(N) the topology from the Cantor space {0, 1}N . It turns out that F in is dense in P(N), and therefore the set {μ(A) : A ∈ F in} is dense in rng(μ). We will use the notation [1, n] := {1, 2, . . . , n}. A subset of a complete metric space is called a Cantor set if it is non-empty compact and dense-in-itself. We are interested in the following probl
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