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  "body": "\n## Statement\n\nLet $I$ be a real [[interval|0Q1D85B8G6]].\nFor each integer $n \\ge 1$, let $M_n \\colon I^n \\to \\R$ be a function.\nAssume that the sequence $(M_n)$ has these four properties for all $n \\ge 1$.\n\n1. $M_n$ is [[continuous|0X3WGX95FE]] and [[strictly increasing|011EJQWWJS]] in each variable.\n2. $M_n$ is symmetric: its value does not change when its arguments are permuted.\n3. $M_n(x, x, \\dots, x) = x$ for all $x \\in I$.\n4. For all $1 \\le k \\le n$ and $x_1, \\dots, x_n \\in I$, with $m = M_k(x_1, \\dots, x_k)$, one has $M_n(x_1, \\dots, x_k, x_{k+1}, \\dots, x_n) = M_n(m, \\dots, m, x_{k+1}, \\dots, x_n)$, where $m$ is repeated $k$ times.\n\nThen there is a continuous strictly monotone function $\\varphi \\colon I \\to \\R$ such that, for all $n$, $M_n$ is the [[quasi-arithmetic mean|SG5C5TVGRB]] generated by $\\varphi$:\n$$M_n(x_1, \\dots, x_n) = \\varphi^{-1}\\left( \\frac{1}{n} \\sum_{i=1}^{n} \\varphi(x_i) \\right).$$\nConversely, every quasi-arithmetic mean has properties 1 to 4.\n\n## Remarks\n\nThe converse is a direct check: property 4 holds because replacing $x_1, \\dots, x_k$ by $m$ does not change the sum $\\sum \\varphi(x_i)$.\nFor the main direction, one fixes two points $a \u003c b$ of $I$ and defines $\\varphi$ on the numbers $M_n(a, \\dots, a, b, \\dots, b)$ with $k$ copies of $b$ out of $n$ as $k/n$.\nProperties 1 to 4 show that this is consistent and strictly increasing on a dense set of points, and continuity extends it.\nThe theorem was proved independently by Kolmogorov and by Nagumo in 1930.\nStrict monotonicity cannot be weakened to monotonicity: $M_n(x_1, \\dots, x_n) = \\min_i x_i$ is continuous, increasing in each variable and has properties 2 to 4, but it is not a quasi-arithmetic mean.\n\n## Explanations\n\n### L1 human\n\nThere are many ways to average numbers, and this theorem explains why most of them share one shape.\nA good average should not care about the order of the numbers, should return $x$ when all numbers equal $x$, and should grow when one of the numbers grows.\nIt should also allow replacing a group of numbers by their own average without changing the total result.\nAny average with these properties is the ordinary average computed on some hidden scale $\\varphi$.\n\n### L2 human\n\nThe theorem turns a question about means into a [[functional equation|SRAN4EKWS0]] for the generator $\\varphi$, and the solution is unique up to $\\varphi \\mapsto a \\varphi + b$.\nIt is used to justify the choice of particular means in statistics, economics and information theory.\nFor example, adding the requirement $M_n(x_1 + c, \\dots, x_n + c) = M_n(x_1, \\dots, x_n) + c$ narrows the generators to $\\varphi(x) = x$ and $\\varphi(x) = e^{ax}$ up to affine changes.\nThat narrowing is the route from quasi-arithmetic means to Rényi entropies.\n",
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