Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0305
T. Filipczak, G. Horbaczewska
In spite of the Lebesgue density theorem, there is a positive δ such that, for every measurable set A⊂ℝ with λ(A)>0 and λ(ℝA)>0, there is a point at which both the lower densities of A and of the complement of A are at least δ. The problem of determining the supremum δH of possible values of this δ was studied by V. I. Kolyada, A. Szenes and others, and it was solved by O. Kurka. Lower density of A at x is defined as a lower limit of λ(A∩[x-h,x+h])/2h. Replacing λ(A∩[x-h,x+h])/2h by λ(A∩[x-tn,x+tn])/2tn for a fixed decreasing sequence ⟨t⟩ tending to zero, we obtain a definition of the constant δ⟨t⟩. In our paper we look for an upper bound of all such constants.
尽管有勒贝格密度定理，但存在一个正δ，使得对于每一个可测集合a∧λ(a)>和λ(a a)>0的∈a，存在一个点，使得a的低密度和a的补密度都至少为δ。V. I. Kolyada, A. Szenes等人研究了δ h可能值的最大值的确定问题，O. Kurka解决了这个问题。A在x处的低密度定义为λ(A∩[x-h,x+h])/2h的下限。对于一个趋于零的固定递减序列⟨t⟩，用λ(A∩[x-h,x+h])/ 2tn替换λ(A∩[x-tn,x+tn])/2tn，我们得到常数δ⟨t⟩的定义。在本文中，我们寻找所有这些常数的上界。
{"title":"EXCEPTIONAL POINTS FOR DENSITIES GENERATED BY SEQUENCES","authors":"T. Filipczak, G. Horbaczewska","doi":"10.14321/REALANALEXCH.46.2.0305","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0305","url":null,"abstract":"In spite of the Lebesgue density theorem, there is a positive δ such that, for every measurable set A⊂ℝ with λ(A)>0 and λ(ℝA)>0, there is a point at which both the lower densities of A and of the complement of A are at least δ. The problem of determining the supremum δH of possible values of this δ was studied by V. I. Kolyada, A. Szenes and others, and it was solved by O. Kurka. Lower density of A at x is defined as a lower limit of λ(A∩[x-h,x+h])/2h. Replacing λ(A∩[x-h,x+h])/2h by λ(A∩[x-tn,x+tn])/2tn for a fixed decreasing sequence ⟨t⟩ tending to zero, we obtain a definition of the constant δ⟨t⟩. In our paper we look for an upper bound of all such constants.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43485581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0269
R. Gardner
This obituary in memory of Washek F. Pfeffer describes his life and character. It also touches on his work in topological measure theory, but other aspects of his research are treated in the adjoining articles by De Pauw, Gruenhage, and Moonens.
{"title":"WASHEK PFEFFER November 14, 1936 — January 3, 2021","authors":"R. Gardner","doi":"10.14321/REALANALEXCH.46.2.0269","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0269","url":null,"abstract":"This obituary in memory of Washek F. Pfeffer describes his life and character. It also touches on his work in topological measure theory, but other aspects of his research are treated in the adjoining articles by De Pauw, Gruenhage, and Moonens.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49107275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0505
Rodolfo Erodias Maza, Sergio Rosales Canoy
A function F:[a,b]→X is said to be an SL function if it satisfies the Strong Lusin (SL) condition given as follows: for every θ-nbd U and a set E⊂[a,b] of measure zero, there exists a gauge δ such that for every δ-fine partial partition D={([xi-1,xi],ti):1≤i≤n} of [a,b] with ti∈E, there exist θ-nbds U1,U2,…,Un such that ∑i=1nUi⊆V and F(xi)-F(xi-1)∈Ui for each i=1,2,…,n. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.
{"title":"ON THE SL-INTEGRAL OF LCTVS-VALUED FUNCTIONS","authors":"Rodolfo Erodias Maza, Sergio Rosales Canoy","doi":"10.14321/REALANALEXCH.46.2.0505","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0505","url":null,"abstract":"A function F:[a,b]→X is said to be an SL function if it satisfies the Strong Lusin (SL) condition given as follows: for every θ-nbd U and a set E⊂[a,b] of measure zero, there exists a gauge δ such that for every δ-fine partial partition D={([xi-1,xi],ti):1≤i≤n} of [a,b] with ti∈E, there exist θ-nbds U1,U2,…,Un such that ∑i=1nUi⊆V and F(xi)-F(xi-1)∈Ui for each i=1,2,…,n. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46518583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/realanalexch.46.2.0279
Thierry de Pauw
{"title":"COMMENTS ON WASHEK PFEFFER’S CONTRIBUTIONS TO INTEGRATION THEORY","authors":"Thierry de Pauw","doi":"10.14321/realanalexch.46.2.0279","DOIUrl":"https://doi.org/10.14321/realanalexch.46.2.0279","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"21 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66983815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0289
L. Moonens
We here briefly describe the importance of Washek F. Pfeffer’s three books on Riemann-type integration and its applications.
我们在这里简要介绍了Washek F.Pfeffer关于黎曼型积分及其应用的三本书的重要性。
{"title":"WASHEK PFEFFER’S BOOKS ON RIEMANN-TYPE INTEGRATION","authors":"L. Moonens","doi":"10.14321/REALANALEXCH.46.2.0289","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0289","url":null,"abstract":"We here briefly describe the importance of Washek F. Pfeffer’s three books on Riemann-type integration and its applications.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44747853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0367
Miroslav Repický
In [13] we gave combinatorial characterizations of non(P) of spaces expressing non-distinguishability of some ideal convergences and semi-convergences of sequences of continuous functions. In the present paper we study three of these invariants: non((I,JQN)-space), none((I,≤KJQN)-space), and none(w(I,JQN)-space). We study them in connection with partial orderings of ωω restricted to relations between I-to-one functions and J-to-one functions. In particular we prove that none(w(I,JQN)-space)≤b for every capacitous ideal J on ω. This generalizes the same result of Kwela for ideals J contained in an Fσ-ideal. If J is a capacitous P-ideal, then non((I,JQN)-space)=none((I,≤KJQN)-space)=b for every ideal I⊆J and none(w(I,JQN)-space)=b for every ideal I below J in the Katĕtov partial quasi-ordering of ideals.
{"title":"SPACES NOT DISTINGUISHING IDEAL CONVERGENCES OF REAL-VALUED FUNCTIONS, II","authors":"Miroslav Repický","doi":"10.14321/REALANALEXCH.46.2.0367","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0367","url":null,"abstract":"In [13] we gave combinatorial characterizations of non(P) of spaces expressing non-distinguishability of some ideal convergences and semi-convergences of sequences of continuous functions. In the present paper we study three of these invariants: non((I,JQN)-space), none((I,≤KJQN)-space), and none(w(I,JQN)-space). We study them in connection with partial orderings of ωω restricted to relations between I-to-one functions and J-to-one functions. In particular we prove that none(w(I,JQN)-space)≤b for every capacitous ideal J on ω. This generalizes the same result of Kwela for ideals J contained in an Fσ-ideal. If J is a capacitous P-ideal, then non((I,JQN)-space)=none((I,≤KJQN)-space)=b for every ideal I⊆J and none(w(I,JQN)-space)=b for every ideal I below J in the Katĕtov partial quasi-ordering of ideals.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45031448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0345
Maxim J. Goldberg, Seonja Kim
Let X be a complete positive σ–finite measure space and {At}t≥0 be a symmetric diffusion semigroup of contraction operators on Lp(X). We prove that for 1
{"title":"AN EXPLICIT CHARACTERIZATION OF THE DOMAIN OF THE INFINITESIMAL GENERATOR OF A SYMMETRIC DIFFUSION SEMIGROUP ON LP OF A COMPLETE POSITIVE SIGMA-FINITE MEASURE SPACE","authors":"Maxim J. Goldberg, Seonja Kim","doi":"10.14321/REALANALEXCH.46.2.0345","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0345","url":null,"abstract":"Let X be a complete positive σ–finite measure space and {At}t≥0 be a symmetric diffusion semigroup of contraction operators on Lp(X). We prove that for 1<p<∞, the domain of the infinitesimal generator of the semigroup is precisely the space ∫01Ash ds:h∈Lp(X). We also establish that for 1<p<∞, the function spaces 2n-1∫02-(n-1)Ash ds|h∈Lp(X) are equal for every n∈ℤ.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47828414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0495
Marek Bienias, S. Gła̧b
The paper presents a generalization of the density point’s notion to the ideal-convergence framework. For an ideal I⊆P(ℕ) (with Fin⊆I), Lebesgue measurable set A⊆ℝ we introduce a definition of a density point of A with respect to I; we prove that the classical approach fits into this generalization (Theorem 4); we construct a family of Cantorlike sets showing that Lebesgue Density Theorem cannot be maximally improved in this direction (Theorem 8).
{"title":"LEBESGUE DENSITY AND STATISTICAL CONVERGENCE","authors":"Marek Bienias, S. Gła̧b","doi":"10.14321/REALANALEXCH.46.2.0495","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0495","url":null,"abstract":"The paper presents a generalization of the density point’s notion to the ideal-convergence framework. For an ideal I⊆P(ℕ) (with Fin⊆I), Lebesgue measurable set A⊆ℝ we introduce a definition of a density point of A with respect to I; we prove that the classical approach fits into this generalization (Theorem 4); we construct a family of Cantorlike sets showing that Lebesgue Density Theorem cannot be maximally improved in this direction (Theorem 8).","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48233914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0319
H. Gaebler
There are three main contributions in this work. First, the proof that every stabilized asymptotic-l1 Banach space has the Property of Lebesgue is generalized to the coordinate-free case. Second, the proof that every Banach space with the Property of Lebesgue has a unique l1 spreading model is generalized to cover a particular class of asymptotic models. Third, a characterization of the Property of Lebesgue is derived that applies to those Banach spaces with bases that admit in a strong sense favorable block bases. These results are significant because they demonstrate not only the efficacy of characterizing the Property of Lebesgue in terms of a connection between the local and the global asymptotic structures of certain Banach spaces, but also the possibility of finding a more general characterization in similar terms.
{"title":"TOWARDS A CHARACTERIZATION OF THE PROPERTY OF LEBESGUE","authors":"H. Gaebler","doi":"10.14321/REALANALEXCH.46.2.0319","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0319","url":null,"abstract":"There are three main contributions in this work. First, the proof that every stabilized asymptotic-l1 Banach space has the Property of Lebesgue is generalized to the coordinate-free case. Second, the proof that every Banach space with the Property of Lebesgue has a unique l1 spreading model is generalized to cover a particular class of asymptotic models. Third, a characterization of the Property of Lebesgue is derived that applies to those Banach spaces with bases that admit in a strong sense favorable block bases. These results are significant because they demonstrate not only the efficacy of characterizing the Property of Lebesgue in terms of a connection between the local and the global asymptotic structures of certain Banach spaces, but also the possibility of finding a more general characterization in similar terms.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48634851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0485
David Rodríguez
In this paper we present a proof of the following statement: there is a C∞ function f on ℝn such that any other C∞ function on ℝn is the uniform limit, on the compact subsets of ℝn, of translations of f by natural numbers. This is a real version of the well-known Birkhoff’s result on the existence of a function with a similar property in the space of entire functions. Afterwards, we show that the technique used in our proof allows us to create 2ℵ0 linearly independent real C∞ universal functions. We also demonstrate that we may even obtain real analytic universal functions (in the sense of translations) by using Whitney’s Approximation Theorem.
{"title":"ON REAL UNIVERSALITY IN THE BIRKHOFF SENSE","authors":"David Rodríguez","doi":"10.14321/REALANALEXCH.46.2.0485","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0485","url":null,"abstract":"In this paper we present a proof of the following statement: there is a C∞ function f on ℝn such that any other C∞ function on ℝn is the uniform limit, on the compact subsets of ℝn, of translations of f by natural numbers. This is a real version of the well-known Birkhoff’s result on the existence of a function with a similar property in the space of entire functions. Afterwards, we show that the technique used in our proof allows us to create 2ℵ0 linearly independent real C∞ universal functions. We also demonstrate that we may even obtain real analytic universal functions (in the sense of translations) by using Whitney’s Approximation Theorem.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48443021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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