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  "body": "\n## Statement\n\nLet $f \\colon \\mathbb{R}^n \\times \\mathbb{R}^m \\to [0, \\infty]$ be Lebesgue measurable.\nThen $x \\mapsto \\int_{\\mathbb{R}^m} f(x, y) \\, dy$ and $y \\mapsto \\int_{\\mathbb{R}^n} f(x, y) \\, dx$ are measurable functions with values in $[0, \\infty]$, and\n$$\\int_{\\mathbb{R}^n \\times \\mathbb{R}^m} f \\, d(x, y) = \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^m} f(x, y) \\, dy \\, dx = \\int_{\\mathbb{R}^m} \\int_{\\mathbb{R}^n} f(x, y) \\, dx \\, dy.$$\nThe three values may be $+\\infty$.\n\n## Remarks\n\nTonelli's theorem has the same conclusion as [[Fubini's theorem|0R0T1VWH5X]]. It replaces integrability of $|f|$ by the hypothesis $f \\ge 0$.\nIn practice, one applies Tonelli's theorem to $|f|$ to check that $\\int |f| \u003c \\infty$, and then applies Fubini's theorem to $f$.\nMore 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 $a_{xy} \\ge 0$.\nLeonida Tonelli published this variation of Fubini's theorem in 1909.\n\n## Explanations\n\n### L1 human\n\nFor a function that is never negative, the order of integration never matters.\nBoth orders give the same value, even when that value is infinite, because no positive and negative parts can cancel.\n",
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