This paper contains the concept of the generalization of the ap-
proximate continuity. The main result concerns that this continuity is equiv-
alent to continuity with respect to some type density topolog

Loranty, Anna

English

Repozytorium Uniwersytetu Łódzkiego (University of Lodz Repository)

Folia Mathematica Acta Universitatis LodziensisVol. 59–65, No., pp.ON THE GENERALIZATION OF THE APPROXIMATECONTINUITYANNA LORANTY‡c© 2004 for University of  Lo´dz´ PressAbstract. This paper contains the concept of the generalization of the ap-proximate continuity. The main result concerns that this continuity is equiv-alent to continuity with respect to some type density topology.Let l be the standard Lebesgue measure in the real line R, and L bethe σ–algebra of Lebesgue measurable sets of R. By N we shall denotethe set of all positive integers and by S the family of all unbounded andnondecreasing sequences of positive numbers. We shall denote a sequence{sn}n∈N ∈ S by 〈s〉. We shall also write A ∼ B if and only if l(A△B) = 0for measurable sets A,B ⊂ R (where A△B = (A \B) ∪ (B \ A)).Definition 1. (See [3]). We shall say that x ∈ R is a density point of a setA ∈ L with respect to a sequence {sn}n∈N ∈ S (in abr. 〈s〉-density point) iflimn→∞l(A ∩ [x− 1sn, x+ 1sn])2sn= 1.Let 〈s〉 ∈ S and A ∈ L. PuttingΦ〈s〉(A) = {x ∈ R : x is 〈s〉 − density point of A},we have the following result.Theorem 1. (See [3]). For each A,B ∈ L and 〈s〉 ∈ S(1) Φ〈s〉(∅) = ∅, Φ〈s〉(R) = R,(2) Φ〈s〉(A ∩B) = Φ〈s〉(A) ∩ Φ〈s〉(B),(3) if A ⊂ B then Φ〈s〉(A) ⊂ Φ〈s〉(B),(4) if A ∼ B then Φ〈s〉(A) = Φ〈s〉(B),(5) Φ〈s〉(A) ∼ A.‡Faculty of Mathematics, University of  Lo´dz´, Banacha 22 St., 90–238  Lo´dz´, Poland.E-mail: loranta@math.uni.lodz.pl.Key words and phrases: approximate continuity, density point, density topology.AMS subject classiﬁcations: 54A10.5960 ANNA LORANTYWe get that for every 〈s〉 ∈ S the family T〈s〉 = {A ∈ L : A ⊂ φ〈s〉(A)}forms a topology on the real line R (see [3]). Of course, if 〈s〉 = {n}n∈Nthen T〈s〉 is simply the clasical ordinary density topology which is denotedby Td (see [2]) and then for every A ∈ LΦ(A) = {x ∈ R : x is a ordinary density point of the set A}.We also recall the following result presented in [3].Theorem 2. Let 〈s〉 ∈ S. ThenT〈s〉 = Td iff lim infn→∞snsn+1> 0.Let S0 = {〈s〉 ∈ S : lim infn→∞snsn+1= 0}.Definition 2. Let f : R −→ R and 〈s〉 ∈ S. We shall say that f is 〈s〉–approximately continuous at point x0 ∈ R if there exsists a set Ax0 ∈ L suchthat(1) x0 ∈ Φ〈s〉(Ax0), and f(x0) = limx→x0,x∈Ax0f(x).Let us recall the concept of appproximate continuity.Definition 3. Let f : R −→ R and 〈s〉 ∈ S. We shall say that f isapproximately continuous at point x0 ∈ R if there exsists a set Ax0 ∈ Lsuch that(2) x0 ∈ Φ(Ax0), and f(x0) = limx→x0,x∈Ax0f(x).Our further results are well known for approximate continuity (see [6]).We do generalization for 〈s〉-approximate continuity. In fact, by Theorem 2the results are evidently extension of the classical results if 〈s〉 ∈ S \ S0.Theorem 3. Let f : R −→ R be a Lebesgue measurable function and let〈s〉 ∈ S. The function f is 〈s〉-approximately continuous at a point x0 ∈ Rif and only if the following condition is satisfied(3) ∀ε>0 x0 ∈ Φ〈s〉({x ∈ R : | f(x)− f(x0) |< ε}).The following lemma will be useful to prove this theorem. The conditionpresented in this lemma is called the condition (J2) of J. M. Je¸drzejewski(see [5]).Lemma 1. Let 〈s〉 ∈ S. For every decreasing sequence {En}n∈N of Lebesguemeasurable sets and for every point x0 ∈ R such that for every n ∈ N wehave x0 ∈ Φ〈s〉(En), there exists a strictly decreasing sequence {kn}n∈N ofON THE GENERALIZATION OF THE APPROXIMATE CONTINUITY 61positive numbers less then 1 and convergent to 0 such that x0 is 〈s〉–densitypoint of set Ax0 =∞⋃n=1(En \ (x0 − kn, x0 + kn)).Proof. Let 〈s〉 ∈ S and let {En}n∈N be a decreasing sequence of Lebesguemeasurable sets. Let us fix x0 ∈ R such that x0 ∈ Φ〈s〉(En) for each n ∈ N.Let {εn}n∈N be a strictly decreasing sequence of positive numbers conver-gent to 0. Since for every n ∈ N we have x0 ∈ Φ〈s〉(En) then(4) ∀n∈N∃k(n)∈N∀k≥k(n)l(En ∩ [x0 −1sk, x0 +1sk])2sk> 1− εn.We can assume that sequence {k(n)}n∈N is increasing. Let {hn}n∈N be asubsequence of the sequence {sn}n∈N expressed in the form hn = sk(n) foreach n ∈ N . Putting kn = εn ·1hn+1for each n ∈ N , we obtain that {kn}n∈Nis strictly decreasing sequence and limn→∞kn = 0. Let Ax0 =∞⋃n=1(En \ (x0 −kn, x0 + kn)) and fix ε > 0. There exists n0 ∈ N such that 1 − 2εn > 1 − εfor any n > n0. Obviously, there exists k0 ∈ N such that1sk< 1hn0+1forany k > k0. Let k > k0. There exsists n1 > n0 such that1sk∈ [ 1hn1+1, 1hn1].This fact with condition (4) implies thatl(Ax0 ∩ [x0 −1sk, x0 +1sk])2sk≥≥l((En1 ∩ [x0 −1sk, x0 +1sk]) \ (x0 − kn1 , x0 + kn1))2sk≥≥l(En1 ∩ [x0 −1sk, x0 +1sk])2sk−2kn12sk>> 1− εn1 −εn1 ·1hn1+11sk≥ 1− εn1 − εn1 > 1− ε.Finally we have that limk→∞l(Ax0∩[x0−1sk,x0+1sk])2sk= 1, which means that x0 is〈s〉-density point of set Ax0 . Proof of Theorem 3. Let 〈s〉 ∈ S.Necessity. Let x0 ∈ R and suppose that f is 〈s〉-approximately continuousat a point x0 ∈ R. Thus there exists a set Ax0 ∈ L such that x0 ∈ Φ<s>(Ax0)62 ANNA LORANTYand∀ε>0∃δ>0∀x∈Ax0 x ∈ (x0 − δ, x0 + δ) ⇒| f(x0)− f(x) |< ε.Hence∀ε>0∃δ>0{x ∈ R :| f(x0)− f(x) |< ε} ⊃ Ax0 ∩ (x0 − δ, x0 + δ).Sincex0 ∈ Φ〈s〉(Ax0 ∩ (x0 − δ, x0 + δ)) and {x ∈ R :| f(x0)− f(x) |< ε} ∈ Lfor each ε > 0 then∀ε>0 x0 ∈ Φ〈s〉({x ∈ R :| f(x0)− f(x) |< ε}).Sufficiency. We assume that condition (3) holds and put for every n ∈ NEn = {x ∈ R :| f(x0)− f(x) |<1n}.Of course, {En}n∈N is a decreasing sequence of Lebesgue measurable setsand x0 is 〈s〉-density point of set En for every n ∈ N . Lemma 1 impliesthat there exists a strictly decreasing sequence {kn}n∈N of positive numbersconvergent to 0 such that x0 is 〈s〉-density point of a set Ax0 =∞⋃n=1(En \(x0 − kn, x0 + kn)). Obviously, Ax0 ∈ L andf(x0) = limx→x0,x∈Ax0f(x).Indeed, fix ε > 0. There exists n0 ∈ N such that1n< ε for each n > n0. Letδ = 1kn0. Since {kn}n∈N is decreasing so, if x ∈ Ax0 ∩ (x0 − kn0 , x0 + kn0)then x ∈∞⋃n=n0+1(En \ (x0 − kn, x0 + kn)). Consequently, if x ∈ Ax0 ∩ (x0 −δ, x0 + δ) then there exists n1 > n0 such that x ∈ En1, which means that| f(x)− f(x0) |<1n1< ε. And we have that f(x0) = limx→x0,x∈Ax0f(x). Definition 4. Let f : R −→ R and 〈s〉 ∈ S. We shall say that f is 〈s〉-approximately continuous if f is 〈s〉-approximately continuous at every pointx ∈ R.Theorem 4. A function f : R −→ R is Lebesgue measurable if and only ifthere exists a sequence 〈s〉 ∈ S such that f is 〈s〉-approximately continuousa.e.Proof. Necessity. Let f be a Lebesgue measurable function. Then f isapproximately continuous a.e. (see [1]). Hence it follows that f is 〈s〉-approximately continuous a.e. for every 〈s〉 ∈ S.ON THE GENERALIZATION OF THE APPROXIMATE CONTINUITY 63Sufficiency. We suppose that there exists 〈s〉 ∈ S such that f is 〈s〉-approximately continuous a.e. Let 〈s〉 ∈ S and α ∈ R. Put E = {x ∈R : f(x) < α}. Let F be a set of all points x ∈ R such that f is 〈s〉-approximately continuous at point x. Then we have E = (E ∩F ) ∪ (E \ F )and l(E \ F ) = 0. We shall show that F ∩ E belongs to L. Firstly we shallshow that the following condition is fulfilled:(5) ∀x∈E∩F∃Ex∈L(Ex ⊂ E ∩ F ∧ x ∈ Φ〈s〉(Ex)).Indeed, let x0 ∈ E ∩ F. Then we have f(x0) < α and there exists a set Ax0such thatx0 ∈ Φ〈s〉(Ax0) and f(x0) = limx→x0,x∈Ax0f(x).We can also assume that x0 ∈ Ax0. Consequently, there exists δ > 0 suchthat f(x) < α for each x ∈ Ax0 ∩ (x0 − δ, x0 + δ). Let Ex0 = Ax0 ∩ (x0 −δ, x0+δ)∩F. Obviously Ex0 ∈ L and Ex0 ⊂ E∩F.Moreover x0 ∈ Φ〈s〉(Ex0).Indeed, x0 ∈ Φ〈s〉(Ax0 ∩ (x0 − δ, x0 + δ)) andlimn→∞l(F ∩[x− 1sn, x+ 1sn])2sn= limn→∞l(R ∩[x− 1sn, x+ 1sn])2sn= 1,so x0 ∈ Φ〈s〉(F ). From the above and Theorem 1 we have that x0 ∈Φ〈s〉(Ex0). And condition (5) is satisfied.Let B be a Lebesgue measurable kernel of the set E ∩ F. For every x ∈E∩F , by condition (5) we have that l(Ex\B) = 0 and Ex ⊂ (Ex\B)∪B ∼ B.From the above and Theorem 1 we have that Φ〈s〉(Ex) ⊂ Φ〈s〉(B) for everyx ∈ E ∩ F. Moreover, condition (5) implies that E ∩ F ⊂⋃x∈E∩FΦ〈s〉(Ex).Consequently,B ⊂ E ∩ F ⊂⋃x∈E∩FΦ〈s〉(Ex) ⊂ Φ〈s〉(B).Simultaneously l(Φ〈s〉(B) \B) = 0. Thus E ∩ F ∈ L and consequently E isa Lebesgue measurable set. Corollary 1. A function f : R −→ R is approximately continuous a.e. ifand only if there exists a sequence 〈s〉 ∈ S such that f is 〈s〉-approximatelycontinuous a.e.Theorem 5. Let f : R −→ R and 〈s〉 ∈ S. A function f is 〈s〉-approximatelycontinuous if and only if the sets Eα = {x ∈ R : f(x) < α} and Eα = {x ∈R : f(x) > α} belong to topology T〈s〉 for every α ∈ R.64 ANNA LORANTYProof. Let f : R −→ R and 〈s〉 ∈ S.Necessity. We have, by Theorem 4, that Eα, Eα ∈ L for each α ∈ R. Letus fix α ∈ R and let x0 ∈ Eα. Then, by Definition 4, there exists Ax0 ∈ Lsuch thatx0 ∈ Φ〈s〉(Ax0), and f(x0) = limx→x0,x∈Ax0f(x).Since f(x0) < α then there exists δ > 0 such that Ax0∩(x0−δ, x0+δ) ⊂ Eα.Consequently,x0 ∈ Φ〈s〉(Ax0 ∩ (x0 − δ, x0 + δ)) ⊂ Φ〈s〉(Eα).And we obtain that Eα ⊂ Φ〈s〉(Eα), which implies that Eα ∈ T〈s〉 for everyα ∈ R.Similary we can show that Eα ∈ T〈s〉Sufficiency. Let x0 ∈ R, fix ε > 0 and suppose that sets Eα and Eα belongto topology T〈s〉 for every α ∈ R. Consequently, f is Lebesgue measurablefunction and{x ∈ R : f(x0)− ε < f(x) < f(x0) + ε} ∈ T〈s〉.Hence, it follows that x0 is 〈s〉-density point of the set {x ∈ R : | f(x) −f(x0) |< ε}. By Theorem 3 we get, that the function f is 〈s〉-approximatelycontinuous at point x0. It follows that f is 〈s〉-approximately continuous. We have the below theoremTheorem 6. Let f : R −→ R and 〈s〉 ∈ S. A function f is 〈s〉-approximatelycontinuous if and only if f is continuous with respect to topology T〈s〉 (inabr. 〈s〉-continuous function).Let 〈s〉 ∈ S. By C(T〈s〉) we shall denote the family of 〈s〉-continuousfunction. In the paper [4] the following result is presentedTheorem 7. Let 〈s〉, 〈t〉 ∈ S. Then C(T〈s〉) = C(T〈t〉) if and only if T〈s〉 =T〈t〉.From the above and Theorems 2 and 6 we have that for every sequence〈s〉 ∈ S0 there exists a function f and a point x0 ∈ R such that f is 〈s〉-approximately continuous at point x0 and is not approximately continuousat point x0.References[1] A. M. Bruckner, Diﬀerentiation of Real Functions, Lecture Notes in Mathematics 659,Springer–Verlag Berlin 1978.[2] K. Ciesielski, L. Larson, K. Ostaszewski, I-Density Continuous Function, Memoirs ofthe American Mathematical Society, 515, 107 (1994).ON THE GENERALIZATION OF THE APPROXIMATE CONTINUITY 65[3] M. Filipczak, J. Hejduk, On topologies associated with the Lebesgue measure, TatraMountains Publications (submitted).[4] M. Filipczak, T. Filipczak, J. Hejduk, On the comparison of the density type topologies,Atti Sem. Mat. Fis. Univ. Modena (submitted).[5] J. M. Je¸drzejewski, On limit numbers of real functions, Fund. Math. 83 (1973/1974).[6] W. Wilczyn´ski, Density topologies, Chapter 15 in Handbook of Measure Theory, Insti-tute of Mathematics, University of Novi Sad, Elsevier, 2002.

On the generalization of the approximate continuity

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On the generalization of the approximate continuity

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