Measure and integration Measures on groups Theorem

Weil's converse to Haar's theorem

Builds on Standard Borel space Sigma-finite measure Quasi-invariant measure Locally compact group Second-countable space Haar measure

Statement

Let $G$ be a group with a standard Borel structure $\mathcal{B}$ such that the map $G \times G \to G$ , $(x, y) \mapsto x y^{-1}$ , is Borel. Let $\mu$ be a nonzero σ-finite measure on $\mathcal{B}$ that is left quasi-invariant: for every $g \in G$ and $A \in \mathcal{B}$ , $\mu(gA) = 0$ if and only if $\mu(A) = 0$ . Then there is exactly one topology on $G$ with the following two properties.

With it, $G$ is a Hausdorff locally compact group that is second countable.
Its Borel $\sigma$ -algebra is $\mathcal{B}$ .

For this topology, $\mu$ has the same null sets as a left Haar measure of $G$ .

Remarks

Haar's theorem goes from a topology to an invariant measure; this result goes back from an invariant measure to a topology. Weil's original version starts from a group with a left-invariant measure that separates points in a suitable sense. It defines a topology from the measure, called the Weil topology. The completion of the group in this topology is locally compact, and the given measure is essentially a Haar measure on it. The form stated here, for standard Borel groups and quasi-invariant measures, is due to Mackey. The idea is that the sets $\set{g : \mu(gA \,\triangle\, A) < \varepsilon}$ , for $A$ of finite positive measure and $\varepsilon > 0$ , generate the neighbourhoods of the identity: their finite intersections form a base. Example: a Polish group that is not locally compact, such as the additive group of an infinite-dimensional separable Hilbert space, carries no nonzero σ-finite left quasi-invariant Borel measure. This explains why there is no analogue of Lebesgue measure on such a space.

Going deeper

For a student

The theorem is used to show that a group has no Haar-like measure: one checks that it cannot be locally compact. A standard case is an infinite-dimensional Banach space, which is a Polish group under addition but is not locally compact. In representation theory, the result lets one recover the topology of a locally compact group from its Borel structure and its measure class. It connects to quasi-invariant measures: on a second countable locally compact group, every σ-finite left quasi-invariant measure has the same null sets as Haar measure.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Haar measure concept
Locally compact group concept
Quasi-invariant measure concept
Second-countable space concept
Sigma-finite measure concept
Standard Borel space concept
Weil's converse to Haar's theorem statement

Tag	`CMZS6KA033`
URI	`m2://CMZS6KA033`
Kind	statement, theorem
Writing	W0 stub
MSC	28C10, 22D05, 03E15
Status	This copy of the site has no build data. Run `m2 status CMZS6KA033`.
About	Haar measure, Locally compact group
Uses	Standard Borel space, Sigma-finite measure, Quasi-invariant measure, Locally compact group, Second-countable space, Haar measure