I have never seen this (1st page) formulation of Fubini's theorem in the literature. Does anyone know where I can find it? In every calculus book (e.g. Apostol, Courant, etc.) I looked, the authors always assume that the function $f$ is integrable. I could find a single author who states (and proves) Fubini's theorem for non-negative functions. I know there is a measure-theoretic Fubini's theorem for non-negative functions that are not necessarily integrable, namely, Tonelli's theorem. However, I am looking for something elementary, something that does not involve measure theory.
I want to justify equalities similar to the one in this question without invoking theorems from measure theory.
Thank you in advance.