---
tag: 9Q5DPYQ5NP
kind: statement
role: theorem
name: Kolmogorov–Nagumo characterization of quasi-arithmetic means
msc: ["26E60", 39B22]
uses: [SG5C5TVGRB, SRAN4EKWS0, 0Q1D85B8G6, 0X3WGX95FE, 011EJQWWJS]
about: [SG5C5TVGRB]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-arithmetic_mean
---

## Statement

Let $I$ be a real [[interval|0Q1D85B8G6]].
For each integer $n \ge 1$, let $M_n \colon I^n \to \R$ be a function.
Assume that the sequence $(M_n)$ has these four properties for all $n \ge 1$.

1. $M_n$ is [[continuous|0X3WGX95FE]] and [[strictly increasing|011EJQWWJS]] in each variable.
2. $M_n$ is symmetric: its value does not change when its arguments are permuted.
3. $M_n(x, x, \dots, x) = x$ for all $x \in I$.
4. 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.

Then 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 $\varphi$:
$$M_n(x_1, \dots, x_n) = \varphi^{-1}\left( \frac{1}{n} \sum_{i=1}^{n} \varphi(x_i) \right).$$
Conversely, every quasi-arithmetic mean has properties 1 to 4.

## Remarks

The converse is a direct check: property 4 holds because replacing $x_1, \dots, x_k$ by $m$ does not change the sum $\sum \varphi(x_i)$.
For the main direction, one fixes two points $a < b$ 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 $k/n$.
Properties 1 to 4 show that this is consistent and strictly increasing on a dense set of points, and continuity extends it.
The theorem was proved independently by Kolmogorov and by Nagumo in 1930.
Strict 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.

## Explanations

### L1 human

There are many ways to average numbers, and this theorem explains why most of them share one shape.
A 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.
It should also allow replacing a group of numbers by their own average without changing the total result.
Any average with these properties is the ordinary average computed on some hidden scale $\varphi$.

### L2 human

The theorem turns a question about means into a [[functional equation|SRAN4EKWS0]] for the generator $\varphi$, and the solution is unique up to $\varphi \mapsto a \varphi + b$.
It is used to justify the choice of particular means in statistics, economics and information theory.
For 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.
That narrowing is the route from quasi-arithmetic means to Rényi entropies.