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

[letmeknow]

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?

 

[HallsofIvy]

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.)

 

[letmeknow]

So can every function be represented as a Fourier equation?

And one more, are we going to finally say that every function is integrable?

 

[HallsofIvy]

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).