Is Fubini's Theorem Always Valid?

[Jhenrique]

I remember a teacher speaking that the fubini's theorem is valid under certain conditions. Implying that not is valid under others. I didn't understood exactly what this teacher was speaking, but the doubt remains still today. We know that $\frac{\partial^2 f}{\partial y \partial x}$ is always equal to $\frac{\partial^2 f}{\partial x \partial y}$, unless that proofs the contrary. The same way, $\int \int f\;dx dy$ is unconditionally equals to $\int \int f\;dy dx$?

 

[lurflurf]

Neither of those are true in general. A commonly given sufficient condition is that limits can be interchanged if the convergence is uniform. That being many times the conditions are met.