For questions related to the Fubini-Tonelli theorem, a theorem for interchanging integrals.
The Fubini–Tonelli theorem, a combination of Fubini's theorem and Tonelli's theorem, states that if $X$ and $Y$ are $σ$-finite measure spaces, and if $f$ is a measurable function, then $$\int_X \left( \int_Y |f(x, y)| dy \right) dx = \int_Y \left( \int_X |f(x, y)| dx \right) dy = \int_{X \times Y} |f(x, y)| \, d(x, y)$$
If any of the three above integrals are finite, then $$\int_X \left( \int_Y f(x, y) dy \right) dx = \int_Y \left( \int_X f(x, y) dx \right) dy = \int_{X \times Y} f(x, y) \, d(x, y)$$
The advantage of the Fubini–Tonelli over Fubini's theorem is that the repeated integrals of the absolute value of $|f|$ may be easier to study than the double integral. As in Fubini's theorem, the single integrals may fail to be defined on a measure $0$ set.
Source: Wikipedia
