Comments on current projects are welcome

The book “Measure-Theoretic Calculus in Abstract Spaces” has been published.

Mon, 12 Feb 2024 00:00:00 -0500

The book “Measure-Theoretic Calculus in Abstract Spaces: On the Playground of Infinite-Dimensional Spaces” has been published on January 24, 2024, by Birkhäuser, an Imprint of Springer-Nature. The DOI link for the book is

https://doi.org/10.1007/978-3-031-21912-2

This book has been seventeen years in the making. It is a great book with mathematically rigorous derivations and encyclopedic treatment on the fundamental topics of real and functional analysis and measure theory that develops and solidifies measure-theoretic calculus in abstract spaces.

Book publication status report III

Sun, 22 Oct 2023 00:00:00 -0400

The book is very close to be published. After a few rounds of proof checking, the book is in a great shape for publication. Personally, I expect two more rounds of proof checking, which will be quick in turn over time. So, the publication date of December 7, 2023 is a very realistic target date. The publication date had been delayed from February 1 to December 7, 2023 many times over the course of the year. This might have caused readers some anxiety. But, it seems that December 7 is a realistic target date for the publication of the book. In any respect, the book is very close to be published. So, the wait is almost over!

Book publication status report II

Mon, 17 Oct 2022 00:00:00 -0400

The final version of the book was submitted in April 2022, and went through another round of review. In the early September, Birkhäuser editorial staff members notified me that they are happy with the book in the final form and will go ahead to initiate publication. So, it is very close the actual publication date for the book. Stay tuned.

Book publication status report

Mon, 04 Oct 2021 00:00:00 -0400

The book is under contract at Birkhäuser since July 2021, and with a delivery date in July 2022. So stay tuned, and soon the book will be available for purchase.

The final draft version of the book is done.

Thu, 09 Jul 2020 00:00:00 -0400

I have completed the final draft version of the book, and would like to have you send me feedback on the book, the style, typos, or mistakes. In this version, I have sharpened the Itô’s Formula so it has less technicalities. It is the final draft as I won’t be adding any additional content into the book. I will only do editorial changes to the book from this point. It has been hard work and a lot of fun in writing this book. Now, I will try to get it published. Hope that this process is not too painful.

A updated draft version of the book is done.

Thu, 09 Apr 2020 00:00:00 -0400

I have completed an updated draft version of the book, and would like to have you send me feedback on the book, the style, typos, or mistakes. In this version, I have sharpened the Itô’s Formula so it has less technicalities. Now, it works like a charm in finite-dimensional spaces and is applicable to infinite-dimensional spaces as well. Also, the integral depending a parameter section is revamped. Now, the results are much sharper and does allow arbitrary measurable function inside the integrand. Some more results on Riemann-Stieltjes integral are obtained that are used to support the Itô’s Formula proof. These results are important in their own right as well.

The draft version of the book is done.

Wed, 09 Oct 2019 00:00:00 -0400

I have completed the draft version of the book, and would like to have you send me feedback on the book, the style, typos, or mistakes. Please, send your comments to me, it will be appreciated. I will try to incorporate your feedback in the final version of the book. Thank you for your patronage over the years. It has been a long journey and mostly enjoyable doing the writing. I am already onto my next book on dynamical systems. Also, I will work out the optimality guided robust adaptive control research. These tasks are next on my to do list.

I am still working on probability theory.

Wed, 28 Aug 2019 00:00:00 -0400

I am still working on Chapter 14 and have corrected a couple of mistakes. The Law of Large Numbers was not proved corrected. The earlier proof only yields the Weak Law of Large Numbers. Now, I have added the Strong Law of Large Numbers at the end of Section 14.4. The proof for the Central Limit Theorem was not entirely correct. The proof is now done correctly and had to involve the logarithm function on the complex plane. The result has to be done properly via a trip to the theory of manifold. I have add Section 12.10 on manifolds. My treatment of manifold is slightly different from the usual exposure of manifold. I think that my treatment is more general. It allows me to resolve the logarithm function and prove the Jacobi identity for vector fields. The proof of the Central Limit Theorem has been corrected accordingly, which I haven’t fully reviewed yet. I am working on the Chapter 14 for a complete review and hopefully finish the book by the end of this year.

A new update.

Sat, 19 Jan 2019 00:00:00 -0500

While working on Chapter 14, I need some general result on Riemann-Stieltjes integral in general spaces. So, I am adding some basic results (generalized from Bartle (1976) book) into Sections A.3 and 12.9. These added stuff is still under development. But, most importantly, I decided to set the title of the book to “Measure Theoretic Calculus in Abstract Spaces'', which I find is more transparent, relevant, appropriate, and also appealing. So, I think that this is the title of the book then.

Chapter 14 is now taking shape

Sat, 20 Oct 2018 00:00:00 -0400

I have spent much time on Probability Theory, Chapter 14. Now, it is taking shape given my research in the previous chapters. The definition of stochastic process has taken a significant update, now I require a stochastic process to be measurable in the product measure space of the time index measure space and the underlying probability measure space. In Professor Burkholder’s notes, these processes are called progressively measurable. For discrete-time processes, this updated version of definition is equivalent to the old definition. For continuous-time processes, this updated version adds significant constraints on what can be called a stochastic process. My past research had been a bit misguided since I never go into the depth of defining a stochastic process, and just take other researchers’ work at their face value. Now, I thought deep and hard into this problem and decided that this definition is the right way to go forward. I hope that this hiccup is going to be forgiven by all my readers. Currently, I am thinking deep and hard into the definition of Itô integral. Hope that I can make something out of it.

Chapters 12 and 13 are now citable.

Sat, 21 Oct 2017 00:00:00 -0400

I have added in Chapter 12 the final version of Change of Variable Theorem for finite-dimensional Euclidean spaces. I decided against including surface measure in Chapter 12, since I don’t have a general theory. I have proofread Chapter 13, and I added a result that the Fourier series is complete in L2((-π,π],IK). I think that I will leave Chapters 12 and 13 as is now, and I may add to these chapters in the future if I think of something interesting. One topic I definitely want to add is Fourier transform. Another would be Laplace transform. But they may not be fit in Chapter 13. But, my current top priority is Chapter 14, Probability Theory. I want to lay the foundation for my teaching and research in stochastic systems.

The Complete Current Notes Including Fundamental Theorems of Calculus

Mon, 09 Oct 2017 00:00:00 -0400

Here is the complete current notes. Everything up to Section 12.7 are correct, which you can cite. I did have some shuffling of propositions around due to new insight gained in the study. I am thinking about whether to impose the condition that an integral is said to exist if it is absolutely integrable. This is in line with the definition of σ-finite banach space valued measures. If the total variation is unbounded, then the measure is not defined. In the current version, an integral may exist even though it is not absolutely integrable. But, then, none of the usual operations that we took for granted for an integral is guaranteed to be valid. So, I am thinking about this change, which is a major rework of Chapters 11 and 12.

The Complete Current Notes

Fri, 14 Jul 2017 00:00:00 -0400

I just uploaded the entire notes that I have for the Real Analysis Reading Note project. Chapters 12 through 15 are not completed yet, and contains mistakes. Please don’t refer to those results. In Chapter 9, I recently noticed that my existing version on the web and on my desk has the wrong definition of C∞ at x0. When I work on papers and the book, I always have the correct notion of this definition in my mind. I don’t know how come I got the wrong definition in the print. I have corrected this definition on pp. 264 and any related statements that refer to this definition. Luckily, this mistake only affects Chapter 9. So, corrections are not that much. I really suspect that it was done by some wannabe slave owner here in my city. But, I did remember that I thought about this definition when I did the notes. I am not sure. Sometimes I wonder why some fellow researchers showed disgust toward my mathematics skill. The above might be the reason. They read my notes and found mistake(s) like this one! This is why I want to leave Cincinnati. This place is evil.

A section on analytic functions in Chapter 9.

Tue, 06 Oct 2015 00:00:00 -0400

I have added a section on analytic functions at the end of Chapter 9. In this section, we properly give the definition of analytic function on normed linear spaces and then show that the composition of analytic functions is again an analytic function. The definition easily guarantees that the elementary functions in normed linear spaces are analytic. We then continue on to prove the analytic versions of Inverse Function Theorem and Implicit Function Theorem.

Chapter 11 is now complete (1)

Fri, 19 Apr 2013 00:00:00 -0400

I have updated Chapter 11 yet again for the following purposes. Propositions 11.140 and 11.190 and Theorem 11.196 are added for sharper results on the normed linear space of finite Y-valued measures on a measurable space. Propositions 11.186 and 11.187 are added for the generation process of σ-finite topological measure space from sequence of finite topological measure spaces. Proposition 11.185 is new. Propositions 11.137, 11.138, 11.145, and 11.146 are added to introduce the notion of vector measure. I am working on Chapter 12 and will include the new chapter after I complete an iteration on that chapter.

Chapter 11 is updated for σ-finite Banach space valued measures

Tue, 18 Oct 2011 00:00:00 -0400

I updated Chapter 11 again. This time, I introduced the concept of σ-finite Banach space valued measures and added a section on the dual of C(X,Y). Much of the results involving Banach space valued measures have been updated. I am still debating whether to include the section on product measure in Chapter 11 or in Chapter 12. I decided to put this version on my site rather than waiting for my decision on the section on product measure.

Chapter 11 is now complete.

Mon, 10 Jan 2011 00:00:00 -0500

I have completed the revision of Chapter 11. Now the theory does not require that the underlying measure space is complete. This has the advantage of simplicity: one only need to work with Borel measurable sets if the functions to start with are Borel measurable. I have revised the definition of topological measure space to accommodate this change. This has led to a better approximation result in Lp spaces by continuous functions (Proposition 11.131).

Chapter 11 is not yet complete

Sat, 21 Aug 2010 00:00:00 -0400

The new Chapter 11 on measure and integration is not yet complete. I am still working on sections on product spaces, fundamental theorem of calculus, and Lp spaces. I have proof read Chapter 11 in the current version. So, the content are correct, even though incomplete.

Real Analysis: local optimization theory.

Wed, 06 May 2009 00:00:00 -0400

In Chapter 10, I didn’t work out a general sensitivity theorem for optimization with equality constraints. This is because I can’t wait to start working on Chapter 11. Finally, I am getting to the integrals.

The minimum phase property.

Wed, 02 Apr 2008 00:00:00 -0400

The minimum phase property will be necessary and sufficient for the disturbance attenuation problem in model reference control design. The necessity is proven in the paper. The sufficiency part will be constructive. It is easy to see that integrator backstepping methodology can be applied to this class of systems to design disturbance attenuating and stabilizing controllers.