When we have that $f_n(x)$ converges to f(x) uniformally in some interval, then the limit of the integrals of $f_n(x)$ is the integral of the limit function - over that same interval, of course.
A more general theorem for taking the limit inside of the integral is the dominated convergence theorem, which can be applied when the integrand $f_n(x)$ is bounded above by an integrable function.
But what about the case for multiple integrals? Say I have not one, nor two, but ... "n" integrals of some function (so that the integrand is a function of n variables), and all the way outside of the n number of integrals, there is a limit to evaluate - either a limit as epsilon goes to zero, or as n goes to infinity, or some other limit.
How can we take this limit ...all the way inside to the innermost integral?
Is it just repeated applications of one of the above two theorems, or am I missing the point, and that there is some other theorem to apply in the multiple integral situation?
P.S. I would like a non-measure-theoretic answer, as this question came up in advanced calculus / introductory real analysis.
Thanks,
Without measure theory, you will have to move the limit through the integrals one at a time. Which sounds awful.
The measure theory answer isn't too bad here. It says that Fubini's theorem is your friend for multiple integration. It tells you when iterated integration is the same as one $n$-dimensional integral. So if your functions are positive, (or integrate well in absolute value), the iterated integrals collapse to one integral. And you can use dominated convergence again.