# 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

[[!redirects Haar measure]]

[[!redirects Haar measures]]

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[[!redirects haar measures]] ]]>

If x is an object of a category C, one usually says that x is if finite presentation (or compact) if for any direct filtered system (y_i) in C, the canonical map

f : colim_i Hom(x,y_i) -> Hom(x, colim y_i)

is bijective. One usually says x is of finite type if this holds only for direct filtered systems of monomophisms.

However, I am interested in objects x for which f is injective (without additional condition on the direct filtered system). It seems to me that, in the category of (right) modules over a ring, such objects are exactly finitely generated modules. My question is, has this been considered, and is there a name for this property of x?

Thanks in advance

Alain Bruguières ]]>

I changed quasicompact to quasicompact morphism though it is also about quasicompact schemes etc. as before and moved the query box here:

]]>Mike: To accord with terminological conventions, this page should probably be either “quasicompact space” or “quasicompact object.”

Zoran Skoda: I do not know what are the conventions, but it was intentional to look both at quasicompact spaces and quasicompact morphisms (which are according to the dominant point of view in algebraic geometry, more important and basic notion); and aside also for q. objects. Personally I do not understand English-language preference for noun phrases. If one is to choose, quasicompact morphism is the choice.

Toby: By the «Each definition gets its own page.» convention, I'm not even sure that this shouldn't just redirect to compact space or compact object. My impression is that assuming that ’compact’ implies Hausdorff is either (like assuming that ’ring’ implies commutative) restricted to fields where it's a common assumption or to languages (I'm thinking mostly of Bourbaki in French here) other than English. On the other hand, if it's used that way by English-writing algebraic geometers, then I would seem to be wrong (since algebraic geometers often have non-Hausdorff spaces).Zoran Skoda: Convention that ’compact’ includes Hausdorff is very common also among people working predominantly on nice spaces, particularly differetial geometers, differential topologists, people studying metric spaces and so on. But for “paracompact” the situation is more tricky: in literature, even on general topology there are also competing definitions, which are all equivalent for Hausdorff spaces. All my life I bounce in such people; my own education does not assume Hausdorffness, unless it is said in the form “compactum”. Algebraic geometers always say quasi-compact, it has nothing to do with language; but as I say for algebraic geometers the basic notion is quasi-compact. The emphasis of this entry is on the terminology and morphisms (what should be expanded on: I still did not write the deifnitions of quasi-compact MORPHISM in various setups); so redirection won’t work I think. Plus although from my point of view saying quasicompact and compact is the same for spaces; one would never say compact for the scheme; scheme is said to be quasicompact if its underlying space is (quasi)compact.

There is an additional reason for that: one can consider a nonsingular variety over complexes which is quasicompact, and which itself is not compact in complex topology (under GAGA). But in the same considerations it is often useful to have some arguments in Zariski and some in complex topology; one of the reasons for word quasicompact is that sometimes we have the “same” example which we are used to think as of noncompact space but it is (quasi)compact in Zariski topology. When an algebraic geometer thinks of the difference between compact and quasicompact for complex varieties he has that in mind; in more general setups about Hausdorff vs nonHausdorff. In the same time, when talking about objects in derived categories of qcoh sheaves, even algebaric geometers use moreoften term compact than quasicompact; thus redirecting to compact object and saying this is for algebraic geometry won’t do for all the 3 notions in this entry (on the contrary side, nobody says compact morphism as far as I could confirm, but quasicompact morphism).

Toby: Ah, so when you've got both Zariski and complex topologies around, you can easily distinguish the former by the prefix ’quasi’; that's cute. Anyway, perhaps we'll move this to quasicompact morphism if you write mostly about that, but I won't try to move anything for now.