Rate of divergence of Lebesgue integrals.
I have a couple of problems from the book: Mathematics for physics and physicists by W. Appel, which I can not solve.
The problems are:
1) Show that
2) Show that
Related to this is given first the defintion:
which in a slightly less general way, but perhaps more comprehensive way can be written as
A theorem is also stated which says:
Let be a non-negative measurable function on , and let be a measurable function on . Assume that is not integrable and that and are both integrable on any interval where . Then the asymptotic comparison between the functions extend to the integrals in the way
My attempt at a solution:
I have tried to solve 1) by noting that
After writing it like this I should be able to utilize the first part of the theorem.
The next step would then be to show that . However, this does not seem to hold as and this does not tend to 1 as so I cannot conclude that the integrands are asymptotically equivalent.
For problem 2) I have even more problems. In this case the x-dependence is not in the limits of the integral and the theorem cannot be applied directly. I have attempted to rewrite the integral by the use of different variable substitutions, but I cannot obtain the required form.
I have worked quite extensively on both of these problems, but I cannot seem to solve them. They are also the only two problems related to the theorem in the chapter I am studying. Either I do not fully comprehend the theorem, or there is something missing. I am a bit unsure about the definition for equivalence between two functions since, in the book, the exact meaning of this is only properly defined for sequences. However, the definition I give above is the direct generalization of this and should be correct.
I am grateful for any suggestions, or help I can get.