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On the Comparison of the Density Type Topologies Generated by Sequences and by Functions

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On the Comparison of the Density Type Topologies Generated by Sequences and by Functions COMMENTATIONES MATHEMATICAE Vol. 49, No. 2 (2009), 161-170 Małgorzata Filipczak, Tomasz Filipczak On the comparison of the density type topologies generated by sequences and by functions Abstract. In the paper we investigate density type topologies generated by functions f(x) f satisfying condition lim inf x > 0, which are not generated by any sequence. x 0+ → 2000 Mathematics Subject Classification: 54A10, 26A15, 28A05. Key words and phrases: density points, density topology, comparison of topologies. Through the paper we shall use standard notation: ℝ will be the set of real numbers, ℕ the set of positive integers, the family of Lebesgue measurable subsets L of ℝ and E the Lebesgue measure of a measurable set E. By Φd(E) we shall | | denote the set of all Lebesgue density points of measurable set E (i.e. Φd(E) = (x h;x+h) E x ℝ; limh 0+ | − ∩ | = 1 ) and by d the density topology consists of ∈ → 2h T nmeasurable sets satisfying E Φdo(E). For any operators Φ, Ψ: we write ⊂ L → L Φ Ψ if Φ (E) Ψ(E) for every E . If Φ Ψ and Φ = Ψ then we write Φ Ψ. ⊂ ⊂ ∈ L ⊂ 6 We will consider two generalizations of Lebesgue density. First of them, called a density generated by a sequence, was introduced by J. Hejduk and M. Filipczak in [6]. For a convenience we will formulate definitions using decreasing sequences tending to zero, instead of nondecreasing sequences going to infinity. Let be the family of all decreasing sequences tending to zero. We will denote S sequences from by (an) or by a . Let a , E and x ℝ. We shall say S h i h i ∈ S ∈ L ∈ that x ise an a -density point of E (a right-hand a -density point of E) if h i h i e e E [x a ; x + a ] E [x; x + a ] lim | ∩ − n n | = 1 ( lim | ∩ n | = 1). n n →∞ 2an →∞ an + By Φ a (E) (Φ (E)) we will denote the set of all a -density (right-hand a - h i a h i h i density) pointsh ofiE. In the same way one may define left-hand a -density point of + h i E and the set Φ−a (E). Evidently, Φ a (E) = Φ a (E) Φ−a (E). h i h i h i ∩ h i 162 On the comparison of the density type topologies In [6] it was proved that Φ a is a lower density operator and the family h i a = E ; E Φ a (E) Th i ∈ L ⊂ h i 1 is a topology containing the density topology d. Moreover for an = ,Φ a = Φd T n h i and a = d. Th i T For any pair of sequences a and b from we denote by a b the decreasing h i h i S h i∪h i sequence consisting of all elements from a and b . It is clear that a b and that h i e h i h i ∪ h i ∈ S e Proposition 1 If Φ a Φ b then Φ a b = Φ a . h i ⊂ h i h i∪h i h i The second type of density we will observe are densities generated by functions. We denote by the family of all functions f : (0; ) (0; ) such that A ∞ → ∞ (A1) lim f(x) = 0, x 0+ → f(x) (A2) lim inf x < , x 0+ ∞ → (A3) f is nondecreasing. Let f . We say that x is a right-hand f-density point of a measurable set E if ∈ A (x; x + h) E lim | \ | = 0. h 0+ f(h) → + By Φf (E) we denote the set of all right-hand f-density points of E. In the same way one may define left-hand f-density points of E and the set Φf−(E). We say that x is an f-density point of E if it is a right and a left-hand f-density point of E. By + Φf (E) we denote the set of all f-density points of E, i.e. Φf (E) = Φf (E) Φf−(E). For any f , the family ∩ ∈ A = E ; E Φ (E) Tf { ∈ L ⊂ f } forms a topology stronger than the natural topology on the real line (see [1, Th. 7] and [4, Th. 1]). Proposition 2 ([4, Prop. 4]) For each f, g , an inclusion f g holds if and only if Φ Φ . ∈ A T ⊂ T f ⊂ g In [2] and [3] it has been shown that properties of f-density operator Φf and f(x) f-density topology f are strictly depended on lim inf x . In our paper we are T x 0+ → f(x) interested in topologies generated by functions f for which lim inf x > 0. ∈ A x 0+ The family of all such functions we will denote by 1. → Topologies generated by functions from 1 andA from 1 have quite different properties (for example they satisfy differentA separationA\A axioms - see [3, Th. 5 and M. Filipczak, T. Filipczak 163 1 Th. 7)]). On the other hand for any function f from properties of f are similar to properties of topologies generated by sequences,A so similar to propertiesT of the density topology (compare [4, Th. 3], [9], [7] and [8]). Moreover Td Proposition 3 ([4, Th. 5]) For any sequence s , the function f defined by a formula h i ∈ S f (x) = s for x (s ; se] n ∈ n+1 n 1 belongs to and Φ s = Φf . A h i Corollary 4 For each a , b , an inclusion a b holds if and only if h i h i ∈ S Th i ⊂ Th i Φ a Φ b . h i ⊂ h i e 1 In [4] we have constructed a function f such that Φf = Φ s for each ∈ A 6 h i s . We will remain the definition of that function, omitting the proof, because weh i will∈ S construct a similar one in Theorem 11. e Example 5 ([4, Th. 6]) Let us define sequences w = (2, 2, 3, 3, 3, 4, 4, 4, 4,...), h i r = (1, 2, 1, 2, 3, 1, 2, 3, 4,...), h i an 1 − a0 = 1, an = 2 for n 1, wn ­ an 1 bn = anwn = − for n 1. wn ­ Evidently an 1 bn lim − = lim = . n n →∞ bn →∞ an ∞ The function f defined by a formula an 1 for x (bn; an 1], f(x) = − − b r for x ∈ (a ; b ]. n n ∈ n n 1 belongs to and Φf = Φ s for each s . A 6 h i h i ∈ S The function constructed above proves that the family ; f 1 is big- e Tf ∈ A ger than s ; s . We will show that there exist continuum topologies from Th i h i ∈ S n1 o f ; f s ; s and that for any pair of sequences a , b satis- T ∈ A \ Th i he i ∈ S h i h i ∈ S 1 fying a bnthere is a functiono f such that a f b and f = s Th i Th i e ∈ A Th i T Th i T e6 Th i for s . Seth i ∈ S x e f (x) = f and s (n) = αs (n) α α α for f 1, s and α > 0. Obviously f 1 and s . ∈ A h i ∈ S α ∈ A h αi ∈ S e e 164 On the comparison of the density type topologies Proposition 6 If Φf = Φ s then Φfα = Φ s . h i h αi Proof It is sufficient to prove that for any measurable set E 0 Φ+ (E) 0 Φ+ (E). fα s ∈ ⇐⇒ ∈ h αi If 0 Φ+ (E) then ∈ fα 1 1 (0; x) E0 (0; αx) E0 x 0+ ∩ α = α | ∩ | → 0, f (x) fα (αx) −→ + 1 + 1 and so 0 Φf α E = Φ s α E . Hence ∈ h i 1 (0; sα (n)) E0 (0; s (n)) E0 n | ∩ | = ∩ α →∞ 0, sα (n) s (n) −→ which gives 0 Φ+ (E). sα ∈ h i + Conversely, if 0 Φ s (E) then ∈ h αi 1 (0; s (n)) E0 (0; sα (n)) E0 n ∩ α = | ∩ | →∞ 0, s (n) sα (n) −→ + 1 + 1 and consequently 0 Φ s α E = Φf α E . Thus ∈ h i x 1 (0; x) E0 α 0; E0 x 0+ | ∩ | = α ∩ α → 0, f (x) x −→ α f α which implies 0 Φ+ (E). ∈ fα ■ Corollary 7 If Φf / Φ s ; s then Φfα / Φ s ; s for any α > 0. ∈ h i h i ∈ S ∈ h i h i ∈ S n o n o e e Theorem 8 If f is the function defined in Example 5 then Φfα Φfβ for any α > β > 1. Proof Since f is nondecreasing, we have f f and Φ Φ . Let (n ) be an α ¬ β fα ⊂ fβ i increasing sequence of positive integers such that rni = 1 for every i. Define ∞ E = ℝ βbni ; βbni √wni . \ i=1 [ It is sufficient to show that 0 Φ (E) Φ (E). ∈ fβ \ fα M. Filipczak, T. Filipczak 165 Without loss of generality, we can assume that for any n βan < bn and βbn < αbn < βbn√wn < an 1. − If x (βbn ; βan 1] for some i then ∈ i i− (0; x) E0 βbni √wni βbni √wni β (1) | ∩ | = = . fβ (x) ¬ x ani 1 √wni f β − If x (βa ; βb ] for some i then ∈ ni ni (0; x) E0 a a 1 (2) | ∩ | < ni = ni = . fβ (x) x rni bni wni f β If x (βap; βbp 1] and ni < p < ni+1 for some i then ∈ − (0; x) E0 a 1 (3) | ∩ | < p .
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