I need to clarify when are the below operations valid. If possible, please link me to the related theorems, where I can find details.
1- Given a double integral \begin{equation} \int_{X}\int_Y f(x,y) \:\mathrm{d}y\:\mathrm{d}x \end{equation} Under what conditions on $f$ the interchange of the order of integration is valid so that \begin{equation} \int_{X}\int_Y f(x,y)\:\mathrm{d}y\:\mathrm{d}x = \int_{Y}\int_X f(x,y)\:\mathrm{d}x\:\mathrm{d}y \end{equation} A related question here, is when we have, instead of a double integral, one integral and infinite sum. For example \begin{equation} \sum_{n=-\infty}^{+\infty} h(n) \int f(x)g(n) \:\mathrm{d}x \end{equation} where $n$ is integer, and $x\in \mathbb{R}$. When is the following valid? \begin{equation} \sum_{n=-\infty}^{+\infty} h(n) \int_X f(x)g(n) \:\mathrm{d}x = \int_X f(x) \left(\sum_{n=-\infty}^{+\infty} h(n)g(n)\right) \:\mathrm{d}x \end{equation}
2- Under what conditions on the integrand, the interchange of the derivative and integral signs is vaild. For example, when can we say that: \begin{equation} \frac{\partial}{\partial t} \left(\iint_S u(x,y,t) \:\mathrm{d}x \:\mathrm{d}y\right)= \iint_S \frac{\partial}{\partial t}u(x,y,t) \:\mathrm{d}x \:\mathrm{d}y \end{equation}