On AP–Henstock–Kurzweil Integrals and Non-Atomic Radon Measure
Round 1
Reviewer 1 Report
Review of mathematics-2165037
Title of the paper: On AP-Henstock-Kurzweil integrals with a non-atomic Radon measure
Authors: Hemanta Kalita, Bipan Hazarika and Tomas Perez Becerra
In the present paper, Authors study an AP-Henstock-Kurzweil type integral on a complete measure metric space X, presenting some of its properties. Also, using locally finite measures, they extend the AP-Henstock-Kurzweil integral to second countable locally compact Hausdorff spaces and establish a Saks-Henstock type Lemma.
The subject is interesting and the proofs are correct, as far as I observed.
The paper can be published after some additions and serious corrections. Our remarks are given below.
Regarding the scientific content
-In the Introduction, Authors have to highlight what are their original contributions and the motivation of their study. They must show what is the novelty brought in this paper, unlike other articles in the literature, and specify what definitions and theorems are new.
-Page 2, in the end of Introduction, before Preliminaries, you have to write about the structure of the paper.
-All over the paper, indicate where all the definitions and results used are taken from (cite them). For exemple, Definitions 2.0.3, 3.0.1 and other.
-Write the notations and the definitions of the following notions: cell function (p. 3), A‾ (p. 7), elementary set (p. 10), fundamental subset (p. 10).
-Write the proofs (at least some summaries) of Theorems 3.1.2, 3.1.3, 3.1.4, 3.1.11.
-State the Vitali Covering Lemma (p. 7).
-State the Cousin`s Lemma (p. 11).
-The references [3], [4], [5], [6], [8], [11], [13], [14], [18], [22] are not cited.
-At p. 12, before DATA AVAILABILITY, a Conclusion section is needed. In the end of Conclusion, some future research and open problems must be presented. This would be very helpful for the interested readers. Here, in the Conclusion section, you can write: In our future research, we will study the converse of Proposition 3.1.15.
-The following papers could be added to the References:
A. Boccuto, A. R. Sambucini The Henstock-Kurzweil integral for functions defined on unbounded intervals and with values in Banach spaces. Acta Mathematica (NITRA) 7, (2004) 3-17.
A. Boccuto - D. Candeloro - A. R. Sambucini A Fubini Theorem in Riesz Spaces for the Kurzweil-Henstock Integral, Journal of Function Spaces and Applications, Volume 9, No. 3 (2011), 283–304. doi:10.1155/2011/158412
Errors in structure, grammar and wording
All over the paper:
-Remove some spaces before the comma or the full stop (for example, at p. 12, in the reference [10]).
-Insert spaces when necessary (for exemple, at p. 11, in the first line of the proof of Prop. 4.1.6, write: Proposition 2.0.4).
-In writing the cited theorems, remove the round brackets (for example, at p. 11, in the first line of the proof of Prop. 4.1.6, replace Proposition(2.0.4) by Proposition 2.0.4).
-R is the set of reals.
Page 1
-Line 1, in the title, write: Henstock and non-atomic.
-In the Abstract, write:
The AP-Henstock-Kurzweil type integral is defined on X, where X is a complete measure metric space. We present some properties of the integral, continuing the study’s use of a Radon measure µr. Finally, using locally finite measures, we extend the AP-Henstock-Kurzweil integral theory to second countable Hausdorff spaces that are locally compact. A Saks-Henstock type Lemma is proved here.
-In Keywords and phrases, write: AP-Henstock-Kurzweil integral. Here, the phrase ”Uniformly strong Lusin condition” does not appear anywhere in the paper. So it must be removed.
-Line 14 of the Introduction, replace ”study” by ”studies”.
-Line 15 of the Introduction, explain ”UAP”.
-Line 16 of the Introduction, write: … investigate convergence theorems …
Page 2
-Line 2, insert a space and remove the full stop in ”et al.”. The same in line 3.
-Line 6, write: … to describe the characterization of …
-Line 7: … et al. mention in their research [2] a type of ...
-Line 8: ... metric space X, using a Radon measure µr and a family …
-Line 10: … R is the set of reals, is enclosed …
-Line 10: … In this paper, we analyse …
-Line 1 of Preliminaries: Let ... with a non-atomic ...
-Line 2 of Preliminaries: … the complete metric spaces or …
-Line 3 of Preliminaries: … interior and the boundary of E by …
-Line 5 of Preliminaries: specify if F is ”a class” or ”the class” of closed non-void subsets of X.
-Line 5 of Preliminaries: Suppose P and Q ….. Why do you need P?
-Line 6 of Preliminaries: remove ”Assume that Q is in F”.
-Def. 2.0.1 is not clear, rephrase it and explain where G is involved. Write: … then F is called a µr –Vitali family.
-Define a fine cover G of E.
-After Def. 2.0.1, write: Consider a fine cover G of E ⊂ X. Recall that a family F of non-void closed subsets of X is a µr-Vitali family if the following Vitali covering theorem is fulfilled:
-Theorem 2.0.2 is not clear, it must be rephrased. Specify if here we have a finite union or a countable union.
-In Def. 2.0.3, write: Consider a measurable set E included in R and ……
-The last text (If there is ……) is not clear. Who is Sxd ? In which set is x ? What is the meaning of (x, [c, d]) ?
-The last line of p. 2, write: Recall ….
-Define S-fine tagged partition.
Page 3
-Line 1, write: Definition 2.0.4 [7, Definition 16.4] A mapping …is called AP-Henstock-Kurzweil integrable if a real number A exists such that …
Here you have to define S(f, P).
-Line 5: A σ-algebra is a collection M of subsets of X satisfying the conditions:
-Lines 7, 8: Replace ”insinuates” by ”implies”.
-Line 9: It is not clear how C appears.
-Line 10: Assume M is the σ-algebra … or Assume M is a σ-algebra … ?
-All over the paper, denote ”μr” by ”μ”.
-Line 14: Remove the phrase: If A, B …., then ….
This is the monotonicity of μ, which clearly follows from the definition of a measure.
-Line 15, write: Recall that a measure μ is called locally finite if for every ….. such that μ(B(x, r)) …
-Line 18, in (2) ……. for every open set V ⸦ X;
-Line 19, in (3), insert a space in ”Vis”. Also here, is the set A arbitrary ?
-Line 20: You wrote two F. Please specify what each represents.
-Line 24, write: The lower derivative LDF is defined similarly. UDF and LDF are studied in [2].
-Line 25: In the definition of ϕ, must be written that Qi is in F, for all i = 1, …, m.
-Proposition 2.0.4 is not correctly formulated.
-In the title of Section 3: ..... IN REGARD TO A RADON MEASURE
-Line 6 from below: In this section, we will discuss the AP- .........respect to a ....
-Line 5 from below, specify F.
-Line 5 from below: An approximate .... is a measurable set ....
-Lines 1-4 from below are not clear and must be rephrased.
Page 4
-Before Definition 3.0.1, specify μ and Q.
-In Definition 3.0.1, write: … a set valued …. is calledap-neighbourhood …
-Line 4, write: subpartition.
-In Definition 3.0.2, specify F and Q.
-In Definition 3.0.2: If there is … at c is 1, then f is called to be approximately …
-In Definition 3.0.2, replace the set function F by μ.
-All over the paper, replace “cell function” by “set function”.
-In Definition 3.0.3: A function … is called …there exists a choice …
-Insert full stop at the end of the relation (1).
-Line 8 of Definition 3.0.3, in ’’which includes”, delete ”which”.
-Remark 3.0.4: Specify Q and define an approximate full coverof Q.
-Line 3 of Remark 3.0.4:delete ”the” and write ”... and the two integrals are equal.”
-In 3.1. Simple properties, line 2, replace ”integral” by ”integrable functions”.
Page 5
-All over the paper, remove the comma after ”we have”.
-Line 8 from above: Consider a gauge ... . Being aware .... that δ is a ...
-Give some ideas for the proofs of Theorems 3.1.2, 3.1.3, 3.1.4.
-In Theorem 3.1.3: Suppose .... If f1(x) = f2(x) for ...
-Theorem 3.1.4: A mapping ... δ on Q so that ... P1 and P2 of Q thar are N-fine.
-Corollary 3.1.5: ... Assume there exists a partition D of Q ... Then f is in ...
-In Definition 3.1.7:
· Suppose ..... and f ... is a ...
· Correct the definition of the map F, that is: F(E) = (AP) ...., for every E in ....
Specify E.
-Proposition 3.1.8: Consider f ... and a partition ... of Q. Then ...
In the first line of the proof, write ” μ (AP)-Henstock-Kurzweil”
All over the paper, use the same notation: μ (AP)-H… or μ-(AP) H… or μAPH…
-Line 2 from below: Now, according …, f is in …. For i = …. m. Then there exists a choice …
Page 6
-Theorem 3.1.9. The map F of Definition 3.1.7 is an additive set function.
Proof. The proof follows from Proposition 3.1.8.
-Rephrase Definition 3.1.10, it is not clear.
-Theorem 3.1.11. Let ... be a set ... on the class of all subsets .... Suppose ... is additive. A mapping f ... is μAPH… integrable on Q …
-Line 15 form below: … Recall that a Radon …, then both are true.
In order to consider …we define the following notions:
-Def. 3.1.12. Let us consider F, a set function …
Denote the set function F with other letter than F.
-Line 3 of Def. 3.1.12, write a “brace” at the end, that is
UAD F(x) = inf { { }}
LAD … = sup { { }}
When … from ∞ and - ∞, then F is called approximate …
Define a point of dispersion…
Page 7
-All over the paper, replace ”cell” by ”set”, ”cells” by ”sets”, ”subcell” by ”subset” and ”subcells” by ”subsets”.
-Theorem 3.1.13. (1) Let ..... be its indefinite ...
Line 5 of the proof: A = ....... LADF∞(x) .... Now, f being μAPH… integrable ….
… < r0r1, …
Since x is a point …of (x-δ, x+δ) \ Sx ….
………….
the set J ….. is a μ-Vitali …
In the proof of Theorem 3.1.13, who is B and Bi ?
-Line 3 from below: where B is a Borel subset of X …
-Line 2 from below: D = ... F∞’AP (x) does not ...... F∞’AP (x) exists ...
-Line 1 from below: Q1 = C U {D} and Q1 is empty ??
-Line 1 from below: ...Let x .... For each ....
Page 7
-All over the paper, replace ”cell” by ”set”, ”cells” by ”sets”, ”subcell” by ”subset” and ”subcells” by ”subsets”.
-Theorem 3.1.13. (1) Let ..... be its indefinite ...
Line 5 of the proof: A = ....... LADF∞(x) .... Now, f being μAPH… integrable ….
… < r0r1, …
Since x is a point …of (x-δ, x+δ) \ Sx ….
………….
the set J ….. is a μ-Vitali …
In the proof of Theorem 3.1.13, who is B and Bi ? And replace f(P) by S(f, P).
-Line 3 from below: where B is a Borel subset of X …
-Line 2 from below: D = ... F∞’AP (x) does not ...... F∞’AP (x) exists ...
-Line 1 from below: Q1 = C U {D} and Q1 is empty ??
-Line 1 from below: ...Let x .... For each ....
Page 8
-Line 1: ... and fk = .... Then by ...
-Line 6: Recalling that on a set ........
....... We find the relation between .... integrable functions and Lebesgue integrable functions ...
-Proposition 3.1.15. Every ..... function ... to μ is … Consequently, (L)… f1 … = (AP) … f1 …
-In the proof of Proposition 3.1.15: Suppose f1 is Lebesgue … Using Vitali …, for … exist the functions f2 and f3 that are … respectively on Q such… Let δ be a N-fine gauge on Q so that f2(t)... and f3(t) … For a N-fine subpartition P = …., i = …. we have
………………..
From (4) and (5), …
-Delete Remark 3.1.16.
-In the title 4. AP-HENSTOCK INTEGRAL WITH RESPECT TO FINITE …
-Line 3 from below, write: locally compact.
-Line 2 from below: … Consider …, where (Xi) …..
Page 9
-Line 1: .... standard Borel measure. A Borel measure ...
-Lines 7 – 12 are not clear and must be reformulated. Who is M ?
-Lines 13 – 31 must be corrected. The notions defined here must be moved at the page 3. Define Q¯ and replace division(s) by partition(s).
-In Definition 4.0.1, specify Q. Write: Let f ...be a function...... Consequently, f is said to be
μLAP Henstock-Kurzweil integrable on Q if ...
We denote the real number A = ...dμ. A is called the APHK integral of f relative to μ on Q.
Definition 4.0.1 must be corrected.
-Line 3 from below: A N-fine ... not simply on Q ..... The set of all functions that are
μLAP Henstock-Kurzweil integrable on Q shall be .....
Remove: We shall sometimes write ....(the whole proposition on the line 1 of page 10).
Page 10
-Line 1: It is easy to see that the integral A is unique.
-Remark 4.0.2 is not clear, it must be reformulated.
-4.1. Within this subsection, we lay out ... The main result here is a Saks-Henstock type Lemma.
-Before Proposition 4.1.1, specify Q, that is: In the sequel, Q is ....
-Proposition 4.1.1. Let Q be ... and f a real valued function defined on Q¯. If ....integrable with the value ...
-In the proof of Proposition 4.1.1: Let f ..., where ... and Y is the ...
Given ..., using Proposition 2.0.4, we can .... Let us define ...., i = 1, 2, ... Then for ..... we have
|S(f, P) – 0| = .... Here you have to write xi instead of x.
-After Proposition 4.1.1, write Proposition 4.1.2. If f1, f2 ..., then for any scalars ..., αf1 + βf2 ...
-Proposition 4.1.2 becomes Proposition 4.1.3. Let .... If the functions ... defined on Q¯ ... almost everywhere in Q¯, then ...
-Proposition 4.1.3 becomes Proposition 4.1.4. Write (Cauchy’s Criterion). Let Q ... function defined on Q¯. ... Then ..P1, P2 of Q that are ..., it holds
-Line 3 from below: ... follows from Definition ... To prove ... be a N-fine ...
Page 11
-Line 1: Let us ... a N-fine ... on Q. Then there is a ... partition Pn of Q which is N-fine for each ...
This implies: ...
This implies ...is a Cauchy sequence ... converges to real number A ... On the condition that P is a N-fine partition on Q, then ...
-Proposition 4.1.4 becomes Proposition 4.1.5. Let ... real valued function defined on Q¯.
-In the proof of Proposition 4.1.5. ...Let P1 .... and P2 ... be N-fine ... and let P3 be a N-fine partition of ... It is very ... P1 ∩ P3 = ϕ, P2 ∩ P3 = ϕ and …, … are N-fine … of Q. Then by the Cauchy’s Criterion
Consequently, we get |S(f, P1) - S(f, P2)| ...
-Proposition 4.1.5 becomes Proposition 4.1.6.
-In the proof, write Cousin’s Lemma.
-Proposition 4.1.6 becomes Proposition 4.1.7. Let .... closed subset of .... Then ... on Q with ...
-In the proof: Let ... from Proposition 2.0.4, ... set U such that ... Let ᴦ be a N-fine gauge on Q ...
The four relations must be corrected.
Page 12
-Theorem 4.1.7 becomes Theorem 4.1.8. Write (Saks-Henstock type Lemma)
Let ... N-fine partition P = ...
-Now, by Proposition 4.1.4 and Remark 4.0.2, ...
-Now, by the ... , finer than ... such that ... we have
-Now, using (6), it results
-After the proof of Theorem 4.1.8, insert the Conclusion section.
-In References, all the titles must be written in italics.
-At [10], write Park.
Author Response
We have replied all the responses of the reviewer.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Referee’s report on
On AP-Henestock-Kurzweil integrals with a non atomic Radon measure
by
Hemanta Kalita; Bipan Hazarika and Tomas Perez Becerra
…..
In this paper, The AP-Henstock-Kurzweil type integral is defined on a complete measure metric space X. In continuing the study's use of the (non-atomic) Radon measure, a Saks-Henstock type lemma is proved. Also, using locally finite measures, extended the AP-Henstock-Kurzweil integral theory to bounded or locally compact, second countable Hausdorf spaces.
- General suggestions.
-A very substantial paper, which has some non-functional parts/proofs and minor spelling mistakes.
- Minor suggestions.
- Page 3, [7, Definition 16.4] A mapping f... You need to write: Definition 2.0.4.
- Page 5, Proof of Theorem 3.1.1. not correct. It is written |S(f1; P2) - A1| <
ε/2; needed |S(f2; P2) - A1| <ε/2.
- Page 6: not a clear proof for a Proposition 3.1.8. δi-fine partition -> δ-fine partition. Why?
- Page 8: Proof. Proof is similar to the [2, Thorem 5.3] so, we omit the proof.
-Page 12: Now by the Proposition4.1.4 and the Remark4.0.2,
- The work is particalary correct.
-Reviewer
Author Response
We have corrected according to the reviewer suggesstions.
Author Response File: Author Response.pdf
Reviewer 3 Report
see attachment
Comments for author File: Comments.pdf
Author Response
We have responded the suggestions of the reviewer.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Review of mathematics-2165037
Title of the paper: On AP-Henstock-Kurzweil integrals with a non-atomic Radon measure
Authors: Hemanta Kalita, Bipan Hazarika and Tomas Perez Becerra
Although the Authors have made all the required changes, there are still some corrections to be made.
The references [8], [11] are not cited.
Page 2
-Line 17: … The construction of the μ-Henstock-Kurzweil integral motivated us …on complete metric spaces with a non-atomic …In this paper, we …Henstock-Kurzweil type, described … covering theorem.
The paper is organized … In Section 3 we … called …Kurzweil integral …
The relationships … are discussed in Proposition 3.1.16.
-Line 4 from below: …The smallest σ-algebra σ(C) containing C, called the σ-algebra generated by C, is the intersection …
Page 3
-Line 3 from above: Then (X, M, μ) …measure space. Suppose U is the Borel σ-algebra of X. Recall that a measure defined on U is called locally finite if for every … such that μ(B(x,r)) ….. where B(x,r) is the open ball of center x and radius r.
-Line 8: (3) ……, for every A ….
In the entire … consider μ a non-atomic ….
Define a μ-set.
-Line 3 of Definition 2.0.1, correct: … such that such that …
Page 4
-Line 10: Specify Sx.
-Line 15: If P is N-fine and [a,b] = ...
-Line 18: ... and the measureis the Borel ...
-Line 8 from below: ... We say that a set ...
-Define an elementary subset.
-Line 1 from below: P = {...
Page 5
-Line 1: … partition of Q can be defined similarly.
-Line 1 of Proposition 2.0.5: If …., then there …
-In Definition 2.0.6: Correct …such that such that …
… called the AP-Henstock-Kurzweil integral of …, and is denoted by A = …
-Line 2 of Section 3: … We consider μ a non-atomic …
Page 6
-Line 1 in Definition 3.0.2, who is Q ? And what means ”μ-valued” set function?
-In Definition 3.0.3: A function … is called …
… number A … there exists a choice …
-Line 3 of Remark 3.0.5, delete ”the” after ”certainly”.
Page 7
-Line 3, in the inequality, have to write A2.
Page 8
-Last line in the proof of Theorem 3.1.4: Hence f is μAPHK …
-In the first line of the proof of Proposition 3.1.7: Since f … is μAP …. on Q, for given …
Now, according to (delete ”the”) Corollary ……. , then there exists a ….
Page 9
-In Definition 3.1.8. We call ..... E of Q (delete ”is”) the indefinite μAPHK integral of …
-Theorem 3.1.9. The map … is an additive set function.
-Theorem 3.1.11. Let …of all subsets of Q.
-Line 8 from below: Additionally, if….correct ”each E subset X”…..Who is B ?
Page 10
-Theorem 3.1.13, in Proof. We prove (1) by contradiction . Remove the space before the full stop.
-Line 9 from below: ... Now, f being (delete ”a”) ...
Page 11
-Lines 2, 3, 4: Who is B and Bi ?
-Line 7: For (2) : Remove a space.
-Line 13 from below: Recall that …
-Theorem 3.1.15: ….. then there exist the functions …
-Proposition 3.1.16: Every …. function …. is ….
Proof. Suppose f1 is Lebesgue (delete ”a”).... Using Vitali .... (without comma) .... Theorem, for .... there exist ....
Page 12
-Line 14 from below: ... X is discrete. A Borel measure …
-Line 7 from below: Who is M?
Page 13
-In 4.1. … In the sequel, Q is a μ-set and …
Page 14
-Line 15: This implies .... sequence that converges ...
-Proposition 4.1.5. Let ..... for every subset ...
Proof. ... and P1 U P3, ... of Q. Then ...
In the lines 4 and 5 from the proof of Proposition 4.1.5, the correct notation is S(f, P).
Page 15
-In the proof of Theorem 4.1.8, line 6: Now, by the ...
-In Conclusions: An AP ..... integral is defined .......
Line 2 from below: is discussed.
Page 16
-Line 1: Lemma is discussed. As a future research topic, we will investigate the validity of the converse of Proposition 3.1.16.
-In reference [1], write the names Boccuto and Sambucini in italics.
Comments for author File: Comments.pdf
Author Response
Response to reviewer 1 in the attached file.
Author Response File: Author Response.pdf
Reviewer 2 Report
Very good paper.
Author Response
Reviewer 2 accepted our previous revised version.
No additional comments
Reviewer 3 Report
The author has made revisions according to the review opinions, which I think is acceptable for publication.
Author Response
Reviewer 3 accepted our previous revised version.
No additional comments
