I Opinion about an introduction of the Lebesgue integral in "Introduction to Hilbert spaces with applications"

[Cauer]

I am an undergraduate student in physics. I wanted to deepen in the theory of Hilbert spaces and I took the book "Introduction to Hilbert spaces with applications" of Debnath and Mikusinski. I am writing this thread to know the opinion of the readers about the method of introducing the Lebesgue integral that one can find in the above-mentioned book. I don't have knowledgement of measure theory. The program in the bachelor's degree in physics is so short in the mathematical aspect! For someone who has worked the book, is "correct" the method for arriving to the Lebesgue integral?

Regards,

 

[fresh_42]

I am an undergraduate student in physics. I wanted to deepen in the theory of Hilbert spaces and I took the book "Introduction to Hilbert spaces with applications" of Debnath and Mikusinski. I am writing this thread to know the opinion of the readers about the method of introducing the Lebesgue integral that one can find in the above-mentioned book. I don't have knowledgement of measure theory. The program in the bachelor's degree in physics is so short in the mathematical aspect! For someone who has worked the book, is "correct" the method for arriving to the Lebesgue integral?

Regards,

While you are waiting for someone who has exactly this one book, you could read
https://www.physicsforums.com/insights/omissions-mathematics-education-gauge-integration/

 

[martinbn]

I am an undergraduate student in physics. I wanted to deepen in the theory of Hilbert spaces and I took the book "Introduction to Hilbert spaces with applications" of Debnath and Mikusinski. I am writing this thread to know the opinion of the readers about the method of introducing the Lebesgue integral that one can find in the above-mentioned book. I don't have knowledgement of measure theory. The program in the bachelor's degree in physics is so short in the mathematical aspect! For someone who has worked the book, is "correct" the method for arriving to the Lebesgue integral?

Regards,

What do you mean by "correct"? Do you think that there are errors in the book?

 

[Cauer]

I mean that I don't know if the development made in the book to arrive to the Lebesgue integral by using step functions and avoiding the theory of measure is correct. Is the approach made by the book authors as valid as the method that starts with the concept of measure and then defines the integral?

 

[martinbn]

I mean that I don't know if the development made in the book to arrive to the Lebesgue integral by using step functions and avoiding the theory of measure is correct. Is the approach made by the book authors as valid as the method that starts with the concept of measure and then defines the integral?

So you think that the book was published even though it may have been incorrect and using a methid that is not valid!

Anyway, no, you dont need to worry. The approach is correct and standard.

 

[mathwonk]

there are two ways to approach lebesgue integrals. one is to treat measure theory and define lebesgue integrals as limits of sums involving measures of complicated sets where a function has values in a given interval. then one proves that in this theory, limits of functions behave well, in the sense that reasonable limits of integrable functions are again integrable, and the integral of the limit is the limit of the integrals.

After the fact, it is clear then that, since limits of integrable functions are lebesgue integrable, we could have just defined lebesgue integrable functions to be such limits. Indeed this approach helped me appreciate lebesgue integration after years of struggling with the complexity of the other definition.

the limit approach is certainly valid and rigorous and is adopted in several very highly regarded books, such as Functional Analysis, by Riesz-Nagy, and Foundations of modern analysis, by Dieudonne'.