What are the conditions to permute a limit and a double integral? For example, if I have to evaluate the limit of this double integral, $$\lim\limits_{j \to \infty}\displaystyle\int _{-j}^j \int _{-j}^{j} \frac{\sin(x)\sin(y)}{xy}\mathrm{d}x\mathrm{d}y,$$ what are the conditions that would give me the right to write that this limit equals to : $$ \displaystyle\int_{-j}^{j}\lim_{j \to \infty}\displaystyle\int _{-j}^{j} \frac{\sin(x)\sin(y)}{xy}\mathrm{d}x\mathrm{d}y\quad?$$
(I'm not supposed to know something about Lebesgue )
Thank you!