Validity of termwise integration
Hello everyone. I just joined the forum today and this is my first post. This looks like a great place with tons of active threads. I hope to contribute in the near future. I am fairly new to math and I hope to broaden my horizons here.
But for now here is my question:
When does the integral (from -∞ to ∞) of an infinite sum equal to the sum of the integral (from -∞ to ∞)? I am looking for a general theorem complete with proof. (Or does one even exist?)
I have seen a few theorems dealing with definite integrals and infinite sums where you need uniform convergence to integrate termwise... but none that deal with infinite sums & integrals from from -∞ to ∞.
If you know of an online source where I could find the theorem/proof could you post it for me please. Or I you have any insight feel free to share.