Integrals (Riemann-Darboux, Riemann, Lebesgue,etc)

In summary, the conversation discusses the different types of integrals (Riemann-Darboux, Riemann-Stieljes, Lebesgue, and Lebesgue-Stieljes) and their applications and limitations. The Lebesgue integral is considered to have the broadest range of integrable functions, but it is not commonly taught until graduate school due to its complexity. The Riemann integral is commonly used in applications and is sufficient for most functions encountered. The Lebesque integral is mainly used in theory and has applications in Fourier transforms. Not all functions are integrable, even with the most general type of integral.
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Homework Statement



My book presents the Riemann-Darboux integral.

It has a small supplemental section on the Riemann integral.

Then a later section on the Riemann-Stieljes integral.

Then a later chapter on the Lebesgue integral.

A supplementary text that I have has a section on the Lebesgue-Stieljes.


My question has a drop of attitude in it; Why am I learning all of these? Will one not do? It appears that the Lebesgue integral (from Wikipedia's say) has the broadest range of integrable functions. Why do they not teach this integral and only this integral?
 
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  • #2
Because it requires some very sophisticated back ground. Generally, it is not taught in the detail that Riemann integration is until graduate school. Also, the ways that one sets up an integral in applications is generally based on the Riemann integral. Finally, for all of the functions that you will meet in applications, the Reimann integral is sufficient. You really need the Lebesque integral and others for theory rather than applications. (Every integrable function has a Fourier transform. In order to be able to say that every Fourier transform is of an integrable function, you have to use Lebesque integral.)
 
  • #3
So can every function be represented as a Fourier equation?

And one more, are we going to finally say that every function is integrable?
 
  • #4
No, I didn't say that. And, no, even with the most general type of integral, the Lebesque integral, there exist non-integrable functions (and even "non-measurable" sets).
 

1. What is the difference between Riemann and Lebesgue integration?

Riemann integration is based on dividing a function into smaller, finite intervals and approximating its area by using rectangles. Lebesgue integration, on the other hand, is based on dividing the function into smaller, infinitesimal intervals and taking the limit as these intervals approach zero. This allows for a more general and powerful approach to integration, as it can handle a wider range of functions and is not limited by the size of the intervals.

2. What is the Riemann-Darboux integral?

The Riemann-Darboux integral is a generalization of the Riemann integral that uses upper and lower sums to approximate the area under a curve. It is defined as the limit of these sums as the size of the intervals approaches zero. This integral is equivalent to the Riemann integral when the function being integrated is continuous, but it can also be used for functions that are not continuous.

3. How do you determine whether a function is Riemann integrable?

A function is Riemann integrable if the upper and lower Riemann sums converge to the same value as the size of the intervals approaches zero. This means that the function must be bounded and have a finite number of discontinuities within the interval of integration. If these conditions are met, the function is considered to be Riemann integrable.

4. What is the fundamental theorem of calculus?

The fundamental theorem of calculus states that differentiation and integration are inverse operations. This means that if a function is integrable on a given interval, its derivative can be found by taking the integral of the function over that interval. This theorem forms the basis of many important mathematical concepts and applications.

5. How is the Lebesgue integral used in real-world applications?

The Lebesgue integral is used in various fields of science and engineering, such as physics, economics, and signal processing. It allows for the integration of more complex and abstract functions, making it a powerful tool for theoretical and applied mathematics. For example, it is used in the Fourier transform, which is essential in many areas of physics and engineering.

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