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  "body": "\n## Statement\n\nLet $(a_{m,n})_{m, n \\in \\N}$ be a [[double sequence|WK5V372TG3]].\n\n1. If $a_{m,n} \\ge 0$ for all $m, n$, then $\\sum_{(m,n) \\in \\N \\times \\N} a_{m,n} = \\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty} a_{m,n}$ as elements of $[0, \\infty]$, where the left side is the supremum of the finite sums.\n2. If $a_{m,n} \\in \\C$ and the family $(a_{m,n})_{(m,n) \\in \\N \\times \\N}$ is [[summable|Z4H7272S9S]], then for each $m$ the series $\\sum_n a_{m,n}$ converges [[absolutely|0W3K3QPEWT]], the series $\\sum_m \\sum_n a_{m,n}$ converges absolutely, and\n$$\\sum_{(m,n) \\in \\N \\times \\N} a_{m,n} = \\sum_{m=0}^{\\infty} \\sum_{n=0}^{\\infty} a_{m,n}.$$\n\n## Remarks\n\nFor part 1, every finite subset of $\\N \\times \\N$ lies in a rectangle, and every iterated partial sum $\\sum_{m \\le M} \\sum_{n \\le N}$ is a finite sum; comparing suprema gives the equality.\nFor part 2, split the real and imaginary parts into positive and negative parts, each bounded by $\\abs{a_{m,n}}$, and apply part 1 to each.\nSummability of a complex family over $\\N \\times \\N$ is equivalent to $\\sum_{m} \\sum_{n} \\abs{a_{m,n}} \u003c \\infty$, by part 1.\nThe statement is the case of the counting measure in [[Tonelli's theorem|067SYD2J4B]] and [[Fubini's theorem|0R0T1VWH5X]].\nExample: for $\\abs{x} \u003c 1$ and $\\abs{y} \u003c 1$, the family $x^m y^n$ is summable, and its sum is $\\frac{1}{1 - x} \\cdot \\frac{1}{1 - y}$.\nWithout summability the conclusion fails: [[interchange of summation order|MZM1AFXY82]] gives a double sequence whose two iterated sums differ.\n\n## Explanations\n\n### L1 human\n\nTo add all the entries of an infinite table, you may add each row first and then add the row totals.\nThis is safe when all entries are nonnegative, or when the sum of the absolute values is finite.\nWithout such a condition, adding row by row can give a different answer than adding column by column.\n\n### L2 human\n\nThe theorem is the discrete form of Fubini's theorem, and it is used constantly with power series and Dirichlet series.\nTypical uses are the [[Cauchy product of absolutely convergent series|0AN28XQ45Z]], the rearrangement of a double power series, and the exchange of two sums in number theory.\nThe nonnegative case needs no hypothesis at all, which makes it the usual first step: one checks $\\sum_m \\sum_n \\abs{a_{m,n}} \u003c \\infty$ and then applies part 2.\n",
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