CLOSURE DENSITY AND DECOMPOSITIONS Milan

CLOSURE DENSITY AND DECOMPOSITIONS Milan Paˇst´eka

Notation. The set of positive integers we shall denote by N. By the symbol d we mean the asymptotic density, D will be the system of all subsets of N having asymptotic density. Buck’s measure density we shall denote as µ and the system of Buck’s measurable sets will be Dµ . For arbitrary sequence {xn } we put A({xn }, S) = {n ∈ N; xn ∈ S}. (n)

Theorem 1. Let M be a non empty set. Suppose that Bn = {Bj ; j = 1, ..., kn } is a system of finite decompositions of M such that for (k) every n = 1, 2, 3, . . . there exists N that each set Bj , k ≤ n is a union of the sets from BN . Then for each finitely additive measure δ:

∞ ∪

Bn → [0, 1]

n=1 (k)

there exists a sequence {xn } of elements of M that A({xn }, Bj ) ∈ Dµ and (k) (k) µ(A({xn }, Bj )) = δ(Bj ), k ∈ N, j = 1, ..., kn . Proof. Let us define the binary relation on M as follows: x ≃ y if and only if the elements x, y belong to the same set from Bn for each n ∈ N. This is a relation of equivalence. Put X = M/ ≃ and x = {y ∈ M; y ≃ x}, x ∈ M. (n)

If we denote Aj

(n)

(n)

= {x; x ∈ Bj }, n ∈ N, 1 ≤ j ≤ kn then En = {Aj ; 1 ≤ (n)

(n)

j ≤ kn } is the system of decompositions of X fulfilling and ∆(Aj ) = δ(Bj ) can be extend to the finitely additive probability measure on the algebra of sets generated by ∪∞ n=1 En . The system of decompositions En , n ∈ N fulfills the conditions (i) and (ii) in [IPT]. The proof is complete. Let x = {xn } be a sequence dense in unit interval [0, 1]. Assume that S ⊂ N. Denote νx∗ (S) = λ(cl({xn ; n ∈ S}), 1

where λ is Lebesque measure on the unit interval and cl is topological closure of set in [0, 1]. For S1 , S2 subsets of N we have {xn ; n ∈ S1 ∪ S2 } = {xn ; n ∈ S1 } ∪ {xn ; n ∈ S2 } and so

νx∗ (S1 ∪ S2 ) + νx∗ (S1 ∩ S2 ) ≤ νx∗ (S1 ) + νx∗ (S2 ).

This yields that Dx the system of sets S that νx∗ (S) + νx∗ (N \ S) = 1 is an algebra of sets and the restriction νx = ν ∗ |Dx is a density defined on Dx . Since {xn } is dense in unit interval we get immediately Proposition 1. For each subinterval I ⊂ [0, 1] the set A({xn }, I) belongs to Dx and νx (A({xn }, I)) = |I|. And this lead to: Proposition 2. For each Jordan measurable set C ⊂ [0, 1] the set A({xn }, C) belongs to Dx and νx (A({xn }, C)) = λ(C). Let us denote ( [ j − 1 j )) ( [ n − 1 ]) (n) Bj = A {xn }, , ,1 , 1 ≤ j < n, Bn(n) = A {xn }, n n n (n)

for n ∈ N. Then Bn = {Bj , 0 ≤ j ≤ n}, n ∈ N forms a system of decompositions of N fulfilling the condition of Theorem 1. Thus we have Theorem 2. There exists such sequence of positive integers {sk } (n) (n) that A({sk }, Bj ) ∈ Dµ and µ(A({sk }, Bj )) = n1 . It is easy to see that for each S ⊂ N and sequence {kn } of positive integers that kn → ∞ for n → ∞ we have ∞ [j − 1 j ] ∩ ∪ cl({xn , n ∈ S}) = , . (1) kn kn (k ) n=1 B

n

j

∩S̸=∅

Moreover for m1 , m2 ∈ N that m1 |m2 the inclusion [j − 1 j ] [j − 1 j ] ∪ ∪ , ⊂ , m2 m2 m1 m1 (m ) (m ) Bj

2

∩S̸=∅

Bj

1

∩S̸=∅

holds and so from upper semi continuity of measure we get: 2

Theorem 3. Let S ⊂ N. Denote by B(S, n) the number of j, 1 ≤ j ≤ n (n) that S ∩ Bj ̸= ∅. If {mn } is such sequence of positive integers that mn |mn+1 , n ∈ N then νx∗ (S) = lim

n→∞

B(S, mn ) . mn

Example 1. If {xn } is a sequence uniformly distributed modulo 1 then every set S ∈ Dx has asymptotic density and νx (S) = d(S). Example 2. If α > 0 is an irrational number and {xn } = {{nα}} then for every m ∈ N, 0 ≤ r < m we have νx∗ (r + (m)) = 1. Applying Theorem 3 and Theorem 6 on page 46 in [PAS] we get: Corollary 1. If {Nk } is such increasing sequence that Nk |Nk+1 , k = 1, 2, 3 . . . and for every m ∈ N there exists k0 that m|Nk , k ≥ k0 and j BN = rj,k + (Nk ), 0 ≤ j < Nk , k ∈ N then νx∗ = µ∗ . (Thus in this case k νx∗ coincides with Buck’s measure density) Example 3. If {Nk } ia sequence fulfilling the condition of Corollary 1 then each positive integer n can be expressed in the form n=

s ∑

cj Nj , 0 ≤ cj