nForum - Search Results Feed (Tag: group) 2020-05-25T21:01:09-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Haar Measure and Haar Integral https://nforum.ncatlab.org/discussion/10969/ 2020-02-23T13:29:04-05:00 2020-02-23T23:47:58-05:00 Dean https://nforum.ncatlab.org/account/2029/ Unfortunately, my edits were mistaken by the spam detector and I cannot edit the Haar measure page. But I think I have something valuable to add. If it's not too much trouble, could someone add these ... Unfortunately, my edits were mistaken by the spam detector and I cannot edit the Haar measure page. But I think I have something valuable to add. If it's not too much trouble, could someone add these edits?

# Haar measure
{: 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.

: https://arxiv.org/abs/math/0606794

[[!redirects Haar measure]]
[[!redirects Haar measures]]
[[!redirects haar measure]]
[[!redirects haar measures]] ]]>
Stabilizer group https://nforum.ncatlab.org/discussion/10042/ 2019-06-15T09:19:32-04:00 2019-06-16T16:24:36-04:00 jesuslop https://nforum.ncatlab.org/account/1486/ Hi, at Stabilizer Group, the notation for action groupoid used (one slash) seems not to fit with the one at action groupoid (two slashes). Not changing myself for the sake of prudence.

Hi, at Stabilizer Group, the notation for action groupoid used (one slash) seems not to fit with the one at action groupoid (two slashes). Not changing myself for the sake of prudence.

]]>
Questions regarding synthetic Lie theory https://nforum.ncatlab.org/discussion/8242/ 2018-01-16T00:20:49-05:00 2018-01-18T17:51:45-05:00 James Francese https://nforum.ncatlab.org/account/1467/ I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the ...

I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the infinitesimal commutator, and that the functor expressed by this operation factors through formal group laws (FGLs) in the usual way. This reveals that Lie groups are FGLs with respect to first-order infinitesimals.

Now I would like to consider a lined topos equipped with higher-order infinitesimals, and develop in this context a modified notion of microlinearity. I have not yet developed the details of this. But does modifying microlinearity in this way, to yield R-modules by exponentiating FGLs with higher-order infinitesimals, sound reasonable? It is worth saying that in general we want certain polynomial identities to hold in the resulting R-modules, e.g. the Jacobian identity.

While FGLs have been thought of in this way (e.g. Didry in , an attempt to extend Lie theory to include Leibniz algebras), I have not found sources discussing modifications of microlinearity to subsume FGLs in the language of SDG. Some suggestive remarks can be found in Nishimura’s work, such as in the introduction of the paper , where the author discusses prolongations of spaces with respect to polynomials algebras as generalizations of Weil algebras. What do you think, nForum?

]]>
Central extensions of mapping class groups from characteristic classes https://nforum.ncatlab.org/discussion/6478/ 2015-02-23T19:37:39-05:00 2015-02-23T19:37:40-05:00 domenico_fiorenza https://nforum.ncatlab.org/account/37/ With Urs Schreiber and Alessandro Valentino we are finalizing a short note on central extensions of mapping class groups from characteristic classes. A preview of the note is available here: Higher ...

With Urs Schreiber and Alessandro Valentino we are finalizing a short note on central extensions of mapping class groups from characteristic classes.

A preview of the note is available here: Higher extensions of diffeomorphism groups (schreiber)

Any comment or criticism is most welcome

]]>
progroup https://nforum.ncatlab.org/discussion/2033/ 2010-10-28T06:31:38-04:00 2010-10-28T20:24:40-04:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Created progroup, with remarks about the equivalence between surjective progroups and prodiscrete localic groups. Why do we have separate pages profinite space and Stone space which do nothing but ...

Created progroup, with remarks about the equivalence between surjective progroups and prodiscrete localic groups.

Why do we have separate pages profinite space and Stone space which do nothing but point to each other? Is there any reason not to merge them?

]]>