Measure and integration Construction of measures Problem

Banach measure problem

Builds on Lebesgue measure Finitely additive measure Rotation Translation

Problem

Let $n \ge 1$ . Is there a function $\mu$ , defined on all subsets of $\R^n$ , with values in $[0, \infty]$ , such that:

$\mu$ is finitely additive: $\mu(A \cup B) = \mu(A) + \mu(B)$ for disjoint $A, B \subseteq \R^n$ .
$\mu$ is invariant under isometries: $\mu(g(A)) = \mu(A)$ for every isometry $g$ of $\R^n$ , such as a translation or a rotation.
$\mu([0, 1]^n) = 1$ .

Remarks

Such a function is a finitely additive measure on the power set. The countably additive version has a negative answer, by the Vitali set. Hausdorff showed with his paradox on the sphere that the answer is negative for $n \ge 3$ . Banach showed in 1923 that the answer is positive for $n = 1$ and $n = 2$ ; such a function can even be chosen to agree with Lebesgue measure on Lebesgue measurable sets. These functions are called Banach measures, and they are not unique. The difference between the dimensions is explained by amenability. The isometry groups of $\R$ and $\R^2$ are amenable, and the rotation group of $\R^3$ contains a free group on two generators.

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Tag	`JXK6NRDH4V`
URI	`m2://JXK6NRDH4V`
Kind	problem
Writing	W0 stub
MSC	28C10, 43A07, 28A12
Status	This copy of the site has no build data. Run `m2 status JXK6NRDH4V`.
About	Lebesgue measure
Uses	Lebesgue measure, Finitely additive measure, Rotation, Translation
Changed	2 November 2026

Banach measure problem

Problem

Remarks

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