Learning resources for geometric intuition in Lebesgue integration

[rick1138]

I am looking for some good materials on Lebesgue integrals, especially anything with a geometric / visual flavor. Any suggestions would be greatly appreciated.

 

[sal]

I am looking for some good materials on Lebesgue integrals, especially anything with a geometric / visual flavor. Any suggestions would be greatly appreciated.

It's not clear to me (or, I suspect, others) what you're aiming at here.

As I recall, roughly speaking, the Lebesgue integral is evaluated by dividing the interval over which it's being integrated into ranges of equal value instead of just putting the interval through a "bread slicer" as is done in the Riemann integral. The sum is then taken over these (typically disconnected) sets on which the function takes particular values, and the limit is taken as the range of values in each section is reduced to zero. (This was a very rough description...)

The reason it's interesting is that many more functions can be Lebesgue-integrated than Riemann-integrated. But -- and it's a big But -- when the Riemann integral exists for a function, the Lebesgue integral is equal to it, and if a function is so badly behaved that you can't Riemann-integrate it, then it's a pretty strange function, or the set over which it's being integrated is perverse.

So, the "meaning" of the Lebesgue integral is identical to the "meaning" of the Riemann integral; only the procedure is different.

With that in mind, what sort of thing are you looking for? Are you looking for illustrations of how a set would be divided up when taking the "Lebesgue measure" of it? Or are you looking for examples of Lebesgue-measurable sets which are not Riemann-measurable, or Lebesgue-integrable functions which are not Riemann-integrable? Or are you looking for general material on Lebesgue theory? The latter can be found in just about any analysis text (e.g., Rudin, "Principles of Mathematical Analysis", or Rudin, "Real and complex analysis", or browse any college or online bookstore looking for analysis texts).

 

[M98Ranger]

Thank you for sharing your interest in learning about geometric Lebesgue integration. It is a fascinating topic that combines both geometry and measure theory. I would be happy to provide some suggestions for resources that may be helpful in your learning journey.

1. "Geometric Measure Theory" by Herbert Federer - This is a classic text that covers various topics related to geometric measure theory, including Lebesgue integration. It is a comprehensive and well-written book that provides a solid foundation in the subject.

2. "Measure, Integral and Probability" by Marek Capinski and Ekkehard Kopp - This book offers an intuitive and visual approach to measure theory and Lebesgue integration. It includes many examples and exercises that help in understanding the concepts.

3. "Real Analysis and Foundations" by Steven G. Krantz - This book covers a wide range of topics in real analysis, including Lebesgue integration. It presents the material in a clear and intuitive manner, with many geometric illustrations to aid in understanding.

4. "An Introduction to Measure Theory" by Terence Tao - This is a concise and well-written introduction to measure theory and Lebesgue integration. It includes many visual aids and examples to help develop an intuition for the subject.

In addition to these resources, there are also many online lectures and videos available on platforms like YouTube, Khan Academy, and Coursera that cover geometric Lebesgue integration. I recommend exploring these as well to supplement your learning.

I hope these suggestions are helpful and wish you all the best in your studies. Happy learning!