# Haar measure

* table of contents

{: toc}

## Idea

If $G$ is a [[topological group]], a _Haar measure_ is a translation-invariant measure on the [[Borel set]]s of $G$. The archetypal example of Haar measure is the [[Lebesgue measure]] on the (additive group underlying) [[cartesian space]] $\mathbb{R}^n$.

## Definition

The proper generality in which to discuss Haar measure is where the topological group $G$ is assumed to be [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]], and from here on we assume this. (For [[topological group]]s, the Hausdorff assumption is rather mild; it is equivalent to the $T_0$ separation condition. See the discussion at [[uniform space]].)

Let $C_c(G)$ denote the vector space of continuous real-valued functionals with compact support on $G$. This is a [[locally convex topological vector space]] where the locally convex structure is specified by the family of seminorms

<latex>\rho_K(f) = \sup_{x \in K} |f(x)|,</latex>

$K$ ranging over compact subsets of $G$. Recall that a [[Radon measure]] on $G$ may be described as a continuous linear functional

<latex>\mu: C_c(G) \to \mathbb{R}</latex>

which is _positive_ in the sense that $\mu(f) \geq 0$ whenever $f \geq 0$. This defines a measure $\hat{\mu}$ on the $\sigma$-algebra of Borel sets in the usual sense of [[measure theory]], where

<latex>\hat{\mu}(B) = sup \{\mu(f): supp(f) = K \subseteq B, \rho_K(f) = 1\}</latex>

By abuse of notation, we generally conflate $\mu$ and $\hat{\mu}$.

A **left Haar measure** on $G$ is a nonzero Radon measure $\mu$ such that

<latex>\mu(g B) = \mu(B)</latex>

for all $g \in G$ and all Borel sets $B$.

### The Haar Integral

Let $G$ be a topological group, and let $\mathbb{C}[G]$, the group ring over $G$. Let $G \text{-Ban}$ be the category of Banach representations of $G$. Objects in $G \text{-Ban}$ are banach spaces $X$ over $\mathbb{C}$ with a continuous action $G \times X \rightarrow X$. Maps in $C$ are bounded, $G$-equivariant maps. (Alternatively, $G \text{-Ban}$ can be viewed as a category of certain $\mathbb{C}[G]$-modules.)

Let $\text{Top}$ be the category of topological spaces, and consider $[G, \mathbb{C}]_{ \text{Top}}$, a Banach representation of $G$ with action $G \times [G, \mathbb{C}]_{ \text{Top}} \rightarrow [G, \mathbb{C}]_{ \text{Top}}$.

We may view $\mathbb{C}$ as a Banach representation of $G$ where $gz = z$ for each $z \in \mathbb{C}$ and each $g \in G$. $\mathbb{C}$ embeds into $[G, \mathbb{C}]_{\text{Top}}$ as constant functions. We may then consider the exact sequence

<latex>0 \rightarrow \mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}} \rightarrow [G, \mathbb{C}]_{\text{Top}}/ \mathbb{C} \rightarrow 0</latex>

A Haar integral on the $G$-representation $[G, \mathbb{C}]_{\text{Top}}$ is a retract $\int_G : [G, \mathbb{C}]_{\text{Top}} \rightarrow \mathbb{C}$ for the injection $\mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}}$. In other words, it is a function $\int_G : [G, \mathbb{C}]_{\text{Top}} \rightarrow \mathbb{C}$ such that

<latex> \int_G (f_1 + f_2) = \int_G f_1 + \int_G f_2 \ \ \ \forall f_1, f_2 \in [G, \mathbb{C}]_{\text{Top}}</latex>

<latex> \int_G a f = a \int_G f \ \ \ \forall f \in [G, \mathbb{C}]_{\text{Top}}, a \in \mathbb{C}</latex>

<latex> \int_G f^g = \int_G f \ \ \ \forall f \in [G, \mathbb{C}]_{\text{Top}}, g \in G</latex>

<latex> \exists C \in \mathbb{R}_{\geq 0 } : \left| \left| \int_G f \right| \right| \leq C \int_G ||f|| \ \ \ \forall [G, \mathbb{C}]_{\text{Top}}</latex>

The last of these requirements, given the others, is equivalent to continuity of $\int_G$.

It is a fundamental theorem, which we will now show, that there is precisely one Haar Measure.

**Remark:** In some sense, we might wish to show that $\text{Ext}^1_{\mathbb{C}[G]}([G, \mathbb{C}]_{\text{Top}}, \mathbb{C})$ vanishes in an appripriate category; this would show that the sequence

<latex>0 \rightarrow \mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}} \rightarrow [G, \mathbb{C}]_{\text{Top}}/ \mathbb{C} \rightarrow 0</latex>

splits by the usual characterization of extensions via $\text{Ext}^1$. On further contemplation, however, it is sufficient only to show that the trivial $G$-representation $\mathbb{C}$ is an injective object in $G \text{-Ban}$. This could be seen as an equivariant Hahn-Banach theorem.

**Proof:** From the remark, it is sufficient to show that $\mathbb{C}$ is an injective object in $G \text{-Ban}$. Take an injection of Banach representations of $G$, $X \rightarrow Y$. Let $f : X \rightarrow \mathbb{C}$ be a map of Banach representations of $G$. By the (usual) Hahn-Banach theorem, there exists a functional $g : Y \rightarrow \mathbb{C}$ extending $f$, though it may lack $G$-invariance.

Consider the subset of all extensions of $f$ to $Y$. Let $S$ be the collection of $G$-invariant compact convex subsets of this set. $S$ contains the convex hull of $G g$, where $g$ is some chosen extension of $f$ to $Y$, so $S$ is nonempty. Using compactness and Zorn's lemma, we may find a minimal element of $S$ in this collection, where $S$ is ordered where $A \leq B$ when $A \subset B$. Call this element $H$. $H$ must be a singleton. If $H$ contains a point which is not extremal then it contains the convex hull of the orbit of that point, which would be a proper $G$-invariant compact convex subset of $H$ (see Krein Milman theorem).

Therefore $H$ is a singleton, and its unique element is a $G$-invariant functional extending $f$.

In particular, since $\mathbb{C}$ has been shown to be injective, the map $\text{Id}_{\mathbb{C}} : \mathbb{C} \rightarrow \mathbb{C}$ lifts along the inclusion

<latex>0 \rightarrow \mathbb{C} \rightarrow [G, \mathbb{C}]_{\text{Top}}</latex>

**Remark:** this alone does not show uniqueness. However, uniqueness is not hard.

**Remark:** by the Riesz-Markov-Kakutani representation theorem, it follows that there is a unique Haar measure on $G$. This result was first proven by Weil. A proof along different lines can be found in these online [notes](http://simonrs.com/HaarMeasure.pdf) by Rubinstein-Salzedo.

### Left and Right Haar Measures that Differ

The left and the right Haar measure may or may not coincide, groups for which they coincide are called **unimodular**.

Consider the matrix subgroup

<latex>

G := \left\{ \left.\, \begin{pmatrix} y & x \\ 0 & 1 \end{pmatrix}\,\right|\, x, y \in \mathbb{R}, y \gt 0 \right\}

</latex>

The left and right invariant measures are, respectively,

<latex>

\mu_L = y^{-2} \,\mathrm{d}x \,\mathrm{d}y,\quad \mu_R = y^{-1} \,\mathrm{d}x \,\mathrm{d}y

</latex>

and so G is not unimodular.

[[Abelian groups]] are obviously unimodular; so are [[compactum|compact]] groups and [[discrete topology|discrete]] groups.

[1]: https://arxiv.org/abs/math/0606794

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]]>I have a very strange question. When people tell me that stacks are abstract nonsense that only algebraic geometers care about (or something similar) I try to point out that stacks occur in most branches of math even if you don’t think of them as stacks (most prominently the stack of vector bundles for differential geometers).

A joke I often make is that there is probably even a stack of measures on a (sufficiently nice) topological space that might interest measure theorists. This question is for people who know some measure theory. Today I was talking about this with an analyst, but the language barrier made it hard to actually verify.

Suppose $X$ is nice enough so that nothing fishy goes on with Borel measures. Then take $Op(X)$ to be the standard site of open sets. Form the category of pairs $(U, \mu)$ where $U$ an open and $\mu$ a Borel measure on $U$. We’ll do something kind of dumb for the arrows and say $(U,\mu)\to (V, \nu)$ is an inclusion $V\subset U$ together with an actual equality of measures $\mu|_V=\nu$.

Just take the forgetful functor and since everything is so rigid this forms a category fibered in sets over $Op(X)$ and is actually seems to be a stack. Now it probably isn’t interesting at all considering you can’t ever have a non-trivial automorphism of $(U,\mu)$. I was wondering if anyone has ever thought of this, or checked this, or come up with something more interesting in a similar vein that I can start using as an example. In theory you could try to use cohomology or something to study measures in this way or talk about “deformations of measures” or something.

The thing that the analyst did say is that measure theorists might care if you could somehow do this up to “bi-Lipschitz equivalence”. We thought about that. The arrows $(U,\mu)\to (V,\nu)$ would then be a bi-Lipshcitz homeomorphism $f:U\to U$ so that $(f_* \mu)|_V=\nu$. It also seems to form a stack (in sets) and moreover you could get lots of automorphisms (for instance take the disk in $\mathbb{R}^2$ and area measure, then any rigid automorphism gives the same measure back) so the stackiness is actually useful. Can anyone else quickly see if this is obviously true or not true for some reason?

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