Tonelli's theorem

Statement

Let $f \colon \mathbb{R}^n \times \mathbb{R}^m \to [0, \infty]$ be Lebesgue measurable. Then $x \mapsto \int_{\mathbb{R}^m} f(x, y) \, dy$ and $y \mapsto \int_{\mathbb{R}^n} f(x, y) \, dx$ are measurable functions with values in $[0, \infty]$, and $$\int_{\mathbb{R}^n \times \mathbb{R}^m} f \, d(x, y) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^m} f(x, y) \, dy \, dx = \int_{\mathbb{R}^m} \int_{\mathbb{R}^n} f(x, y) \, dx \, dy.$$ The three values may be $+\infty$.

Remarks

Tonelli's theorem has the same conclusion as Fubini's theorem. It replaces integrability of $|f|$ by the hypothesis $f \ge 0$. In practice, one applies Tonelli's theorem to $|f|$ to check that $\int |f| < \infty$, and then applies Fubini's theorem to $f$. More generally, the theorem holds for non-negative measurable functions on a product of two $\sigma$-finite measure spaces. For counting measure on $\mathbb{N}$, it says that $\sum_x \sum_y a_{xy} = \sum_y \sum_x a_{xy}$ when all $a_{xy} \ge 0$. Leonida Tonelli published this variation of Fubini's theorem in 1909.

Explanations

L1 human

For a function that is never negative, the order of integration never matters. Both orders give the same value, even when that value is infinite, because no positive and negative parts can cancel.