I need to justify interchanging a limit with a Riemann integral. The most frequently cited justification is that convergence associated with the limit is uniform. However, my applications cannot assume uniform convergence. Instead, a boundedness condition is given. I know how to prove the theorem but writing up a proof is not necessary if the theorem is already well known. I don't see it in my favorite text books so I don't know if it is well known.
The theorem is as follows. Consider a function H(x,t) and two open intervals (a,b) and (A,B). The function has four properties. The first is that it is bounded on (a,b)X(A,B). The second property is that for each t in (A,B), H(x,t) as a function of x is integrable on (a,b). The third property is that for each x in (a,b), H(x,t) as a function of t has a limit as t approaches A. The last property is that this limit, which is a function of x, is integrable on (a,b). The theorem states that the limit as t approaches A can be interchanged with integration in x from a to b. In other words, the limit of the integral is the integral of the limit. Is this theorem well known? Does it have a name?
It's kind of Lebesgue's dominated convergence theorem
Since your domain nice ( compact) Lebegues integral and Riemann integral are equal
Would you speak about Lebesque integrals, and if 'bounded' would mean boundedness of the absolute value, you had simply to apply Lebesgue's dominated convergence theorem.
That, as Alzaki states, Lebesque integral and Riemann integral coincide on compact domains is unfortunately not true. A counterexample is the function which is 1 at all rational numbers and 0 else. It is bounded, has Lebesque integral 0, and the Riemann integral does not exist.
Since your integration interval for x is open, it is not clear whether your Riemann integrals are allowed to be 'improper' at the end (e.g. oscillatory with frequency = infinity at the end point). Your question thus can't be answered as it stands.
Alzaki, I nowhere read 'continuous' in the original question. Am I blind?
The name OSGOOD'S THEOREM would be appropriate.
Osgood, W. F.
On non-uniform convergence and termwise integration of series. (Ueber die ungleichmässige Convergenz und die gliedweise Intergration der Reihen.) (German)
Gött. Nachr. 1896, 288-291 (1896).
Please L.D. Edmonds let us know if your function continuous or not thank you, Hello Prof Mutze, easy please. Yes he didn't say discontinuous as well, but I assume that maybe continuous
The closest hit seems to be Arzela's theorem, which considers Riemann integration but has closed intervals instead of open ones. The appended link seems to be a good reference. My entrance point was the classical volume Functional Analysis by Riesz and Nagy Dover 1990 who discusses Arzela and Osgood theorems of the pre-Lebesque aera.
https://www.math.washington.edu/~morrow/335_13/dominated.pdf
To answer your question, please replace each of my mentions of "integrable" with "sectionally continuous". If it makes things simpler, we can assume "continuous", except for the limit of H as t approaches A, which might have a finite number of discontinuities in the open interval (a,b) but is still bounded.
Here is one more clarification. The function H(x,t), as a function of x, may or may not be defined at the interval endpoints. An example is sin(1/x) which is bounded but not defined at x=0. For such cases, the integral is defined as an improper Riemann integral.
Thank you all for the help, and thank you Prof. Mutze for the link. Arzela's theorem explained in the link will do the job for me. I am doing math for scientists and engineers (specifically, I'm investigating properties of solutions to a diffusion equation), not for mathematicians, so some subtle issues can be overlooked because we engineers usually are a little sloppy with the math. I am surprised though that the proof of Arzela's theorem is so involved. I was expecting a much simpler proof that did not require Lebesgue theory.
Indeed, this theorem is well known and a general for of it can be found in
Benedetto, John J.; Czaja, Wojciech. Integration and modern analysis. Birkhäuser Advanced Texts 2009, sections 3.6 Theorem 3.6.1 ( The interchange of limits and integration), respectively 3.7 Theorem 3.7.12 ( Fubini-Toneli theorem for Lebesgue and Riemann integrable functions). For the boundedness condition you can see the section 3.3. I wrote a review (MR2559803) for this book some years ago.
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