integration theory

[nassima2231]

Hello, what is the principle to define an integral,at the beginning, It was the Riemann integral and the sums of Darboux, etc ... but when we introduced the Lebesgue integral, i don't understand, how do you differentiate between them? and how to see the notion of integral? what are the conditions for it to exist?
(that even if i have another kind of integral i would be convinced), please clarify me.

 

[Ishuda]

Hello, what is the principle to define an integral,at the beginning, It was the Riemann integral and the sums of Darboux, etc ... but when we introduced the Lebesgue integral, i don't understand, how do you differentiate between them? and how to see the notion of integral? what are the conditions for it to exist?
(that even if i have another kind of integral i would be convinced), please clarify me.

As you seem to know the Riemann integral and Darboux integral both give the same value if they exist and if one exists the other also exists. One might, in a sort of non-rigorous (and not completely true but almost) sense say that the basic difference between the Riemann sum and the Darboux is that the partition of the interval in the Riemann integral is for a fixed width which goes to zero and that for the Darboux sum is for a variable width with the maximum width going to zero.

The Lebesgue integral on the other hand, being an extension of the Riemann/Darboux integrals may exist where the other two do not. However if Both the Lebesgue integral and the Riemann/Darboux integrals exist, they are equal to each other [assuming the usual measure for $\displaystyle \mathbf R$ and $\displaystyle \mathbf R$X$\displaystyle \mathbf R$]

 

[nassima2231]

1)Let I be an integral, how we can know that it existe in sens of Lebesgue and not in sens of Riemann?
2)an integral of Lebesgue, has a relation with the derivation or no? i.e if F'=f then integral of f in sens of Lebesgue is F?
how to integrate a function Lebesgue?

 

[Ishuda]

1)Let I be an integral, how we can know that it existe in sens of Lebesgue and not in sens of Riemann?
2)an integral of Lebesgue, has a relation with the derivation or no? i.e if F'=f then integral of f in sens of Lebesgue is F?
how to integrate a function Lebesgue?

SPIT! forum acting up again so I'll try AGAIN

(2) first: If the Riemann or Darboux [R or D] integral exists then all three integrals [Lebesgue (L), Riemann, and Darboux] exist and are the same. Thus, in this case, if F'=f, then then L integral exists and is equal to D and R integral which is F. However, note that formally, the L integral is initially considered only on some set, i.e. is a definite integral, as, initially, the R and D integrals are. So that the L integral between a and b of F' would be f(b)-f(a) as it is for the R & D integrals.

For (1) the classic example of where the L integral exists and the R and D integrals don't is the 'belongs to' function, i.e. the indicator function, on [0,1] for the rationals. That is, let
f(x) = 0 if x is not rational; $\displaystyle 0\, \le\, x\, \le\, 1$
and
f(x) = 1 if x is rational; $\displaystyle 0\, \le\, x\, \le\, 1$
Then, as explained on
http://en.wikipedia.org/wiki/Lebesgue_integration
since the rationals are countable on [0,1], the L integral 'from 0 to 1' of f is zero whereas neither the R nor D integral exists.

 

[pka]

1)Let I be an integral, how we can know that it existe in sens of Lebesgue and not in sens of Riemann?
2)an integral of Lebesgue, has a relation with the derivation or no? i.e if F'=f then integral of f in sens of Lebesgue is F? how to integrate a function Lebesgue?

You are asking for something that this site does not do, teach. Anyone in a PhD program wanting to do research in integration theory takes a series of three courses which answer our question in detail and takes at least a year to complete. Here are three typical books: The Theory of the Integral by S. Saks; Theory of Integrations by T H Hildebrandt; Measure and Integration by S K Berberian. These three cover the information in the order you asked.

Now it will surprise you but “if F'=f then integral of f in sens of Lebesgue is F” is FALSE. It is false for each of the integrals you have named. In other words, there are derivatives of perfectly good functions which are themselves not integrable. It was not until the 1960’s that problem was solved with a new type integral.

Here is a quick bit of history. Newton himself in 1664-86, being an astronomer, was interested in the area swept out by the motion of heavenly orbs. Hence, he first found the relation of area and antiderivative of very well behaved functions. This was the case for almost two hundred years until both Riemann and Lebesgue tried to answer different questions. This study/research area became very active again from 1950-1985.

 

[nassima2231]

for students who know the concept of integration Riemann since the first year, how do you introduce their this notion of Lebesgue?
i'll search more, and if i find any difficulty, i'll ask my questions here. thanks.

 

[Ishuda]

...

Now it will surprise you but “if F'=f then integral of f in sens of Lebesgue is F” is FALSE. ...

As usual pka is correct [in the sense of it is not true for ALL f]. F' does not have to be integrable and, even if it is, it does not have to be identically F under either Riemann, Darboux, or Lebesgue integration. But, if it is under R or D integration, then it is under the other two.

There are many different kinds of integrals and the study area pka is referring to was, in part, heralded by the development of the Henstock–Kurzweil integral, see
http://en.wikipedia.org/wiki/Henstock–Kurzweil_integral
for example.

 

[nassima2231]

Hey, i want to know, for a function f, if the integral of f with respect to the measure of lebesgue it equal to F such as F'=f ?
on the other hand, the integral of f with respect to another measure (for example measure p of probability) is equal to what ??
if the two integrals coincide (with respect to the measure of Lebesgue, and another measure), so where is the difference between these integrals ?

 

[nassima2231]

Hey,
i want to know, for a function f, the integral of Lebesgue is F such as F'=f ?
the integral of f with respect to two measures, for example the measure of Lebesgue and the measure of probability p, are they equal?
if yes, so why we specify the measure before integrating?
if no, how we calculate each one?
please explain me more, please.