nLab
Haar integral

Contents

Contents

Idea

If GG is a locally compact Hausdorff topological group, with C c(G)C_c(G) the algebra of compactly supported continuous functions from C c(G)C_c(G) to \mathbb{R}, a Haar integral G\int_G is a continuous linear map C c(G)C_c(G) \rightarrow \mathbb{R} which is invariant under the aparent action of GG on C c(G)C_c(G). It turns out that there exists a unique such integral, up to a scalar multiple. The Riesz representation theorem then allows one to conclude the existence of a unique Haar measure, which is a GG-invariant Borel measure on GG.

The archetypal example of Haar measure is the Lebesgue measure on the (additive group underlying) cartesian space n\mathbb{R}^n.

Definition

Let GG be a locally compact Hausdorff group. Let C c(G)C_c(G) denote the vector space of continuous real-valued functionals with compact support on GG. This is a locally convex topological vector space where the locally convex structure is specified by the family of seminorms

ρ K(f)=sup xK|f(x)|\rho_K(f) = \sup_{x \in K} |f(x)|

KK ranging over compact subsets of GG. Recall that a Radon measure on GG may be described as a continuous linear functional

G:C c(G)\int_G : C_c(G) \to \mathbb{R}

Such a Radon measure defines a measure μ\mu on the σ\sigma-algebra of Borel sets in the usual sense of measure theory, where

μ(B)=sup{ Gf:supp(f)=KB,ρ K(f)=1}\mu(B) = sup \left\{\int_G f : supp(f) = K \subseteq B, \rho_K(f) = 1 \right\}

In this context, a (left) Haar integral on GG is a nonzero such linear functional G\int_G such that

Gf0 when f0 \int_G f \geq 0 \text{ when } f \geq 0
Gf g= Gf \int_G f^g = \int_G f

for each fC c(G)f \in C_c(G) and each gGg \in G, where f g:Gf^g : G \rightarrow \mathbb{R} sends xx to f(gx)f(gx).

Correspondingly, a (left) Haar measure on GG is a nonzero Radon measure μ\mu such that

μ(gB)=μ(B)\mu(g B) = \mu(B)

for all gGg \in G and all Borel sets BB.

An Analogy with the Finite Case

There is an analogy between Haar measure and scaled-cardinality on a finite group. In fact, the latter is a special case of the former, as we may view a finite group as a discrete topological group. While measure on a (discrete) finite group is subsumed by the notion of Haar measure, it may be of interest to build intuition.

Let G finG_{fin} be a finite group. Let G fin-RepG_{fin} \text{-Rep} be the category of GG-representations of RR-modules, where RR is a ring in which |G fin||G_{fin}| is invertible. This category is equivalent to R[G]R[G]-mod. We can view RR as a trivial G finG_{fin}-representation, where ga=aga = a for each aRa \in R and each gGg \in G.

Let GG be a compactum. Let G-BanG \text{-Ban} be the category of Banach representations of GG. Objects in G-BanG \text{-Ban} are banach spaces XX over \mathbb{R} with a continuous norm preserving action G×XXG \times X \rightarrow X, i.e. ||gx||=||x||gGxX||gx|| = ||x|| \forall g \in G \forall x \in X. Maps in G-BanG \text{-Ban} are short maps which are GG-equivariant. (Alternatively, G-BanG \text{-Ban} can be viewed as a category of certain [G]\mathbb{R}[G]-modules.) We can view \mathbb{R} as a trivial GG-representation, where ga=aga = a for each aa \in \mathbb{R} and each gGg \in G.

Let C(G fin)C(G_{fin}) be the ring of set-maps from G finG_{fin} to RR. GG acts on C(G fin)C(G_{fin}) where f g:G finRf^g : G_{fin} \rightarrow R sends xx to f(gx)f(gx). There is a map of abelian groups G fin:C(G fin)R\int_{G_{fin}} : C(G_{fin}) \rightarrow R sending ff to 1|G| gGf(g)\frac{1}{|G|} \sum_{g \in G} f(g), analogous to the Haar integral. Indeed,

G finf+g= G finf+ G fingf,gC(G fin)\int_{G_{fin}} f + g = \int_{G_{fin}} f + \int_{G_{fin}} g \forall f, g \in C(G_{fin})
G finaf=a G finffC(G fin)aR\int_{G_{fin}} a f = a \int_{G_{fin}} f \forall f \in C(G_{fin}) \forall a \in R
G finf g= G finffC(G fin)gG\int_{G_{fin}} f^g = \int_{G_{fin}} f \forall f \in C(G_{fin}) \forall g \in G
G finf0 when f0fC(G fin)\int_{G_{fin}} f \geq 0 \text{ when } f \geq 0 \forall f \in C(G_{fin})

What corresponds to the Haar measure on GG is simply cardinality (though we must appropriately divide by the cardinality of GG, to get a function μ fin:P(G)R\mu_{fin} : P(G) \rightarrow R from the power set of GG to RR sending SS to |S||G|\frac{|S|}{|G|}).

Using the Haar integral, we may define convolution product: *:C(G×G)C(G)* : C(G \times G) \rightarrow C(G) sending f:G×Gf : G \times G \rightarrow \mathbb{R} to the map GG \rightarrow \mathbb{R} sending hGf(gh 1,h)= hk=gf(h,k)\int_{h \in G} f(gh^{-1}, h) = \int_{hk = g} f(h, k). This is analogous to the map * fin:C(G fin×G fin)C(G fin)*_{fin} : C(G_{fin} \times G_{fin}) \rightarrow C(G_{fin}) sending ff to the map C(G fin)C(G_{fin} ) sending gg to 1|G fin| hk=gf(h,k)\frac{1}{|G_{fin}|} \sum_{h k = g} f(h, k).

In both cases, we get a “bar construction”. For the compact Hausdorff case, we get:

C(G×G)f(g hGf(g,h))*C(G) G \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} C(G \times G) \stackrel{\stackrel{* }{\longrightarrow}}{\stackrel{ f \mapsto \left( g \mapsto \int_{h \in G} f( g, h) \right) }{\longrightarrow}} C(G) \stackrel{\int_G}{\longrightarrow} \mathbb{R}

and for the finite case, we get:

C(G fin×G fin)f(g1|G fin| hGf(g,h))* finC(G fin)f1|G fin| gGf(g) \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} C(G_{fin} \times G_{fin}) \stackrel{\stackrel{*_{fin} }{\longrightarrow}}{\stackrel{ f \mapsto \left( g \mapsto \frac{1}{|G_{fin}|} \sum_{h \in G} f( g, h) \right) }{\longrightarrow}} C(G_{fin}) \stackrel{f \mapsto \frac{1}{|G_{fin}|} \sum_{g \in G} f(g)}{\longrightarrow} \mathbb{R}

Calling this a bar resolution is a slight abuse of terminology; the “monad” involved is actually has no unit, as C(G)C(G) has no unit for the convolution product. However, convolution makes C(G)C(G) an assocciative nonunitial algebra, so that the resolution is still a unitless monad. Hence there are evident face maps without degeneracies.

While C(G)C(G) has no unit, there is a canonical way of adding units to it: take products with [G]\mathbb{R}[G]. Write gGa gδ g\sum_{g \in G} a_g \delta_g for the elements of [G]\mathbb{R}[G]; since δ 1\delta_1 is a formal unit for convolution, we may think of it as a Dirac delta function. Now the bar resolution

C(G×G)[G]f(g hGf(g,h))*C(G)[G] G \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} C(G \times G) \prod \mathbb{R}[G] \stackrel{\stackrel{* }{\longrightarrow}}{\stackrel{ f \mapsto \left( g \mapsto \int_{h \in G} f( g, h) \right) }{\longrightarrow}} C(G) \prod \mathbb{R}[G] \stackrel{\int_G}{\longrightarrow} \mathbb{R}

has degeneracies as well.

Properties

Existence and Uniqueness

Any locally compact Hausdorff topological group GG admits a Haar integral (and therefore Haar measure) that is unique up to scalar multiple. This result was first proven by Weil. A proof can be found in these online notes by Rubinstein-Salzedo. A different, constructive proof, due to E.M. Alfsen, can be found in the article “A simplified and constructive proof of the existence and uniqueness of haar measure”.

We here give a lesser known proof of the existence of the Haar integral, specifically on compact Hausdorff groups GG, which uses convex sets and the Krein Milman theorem instead of measure theory. Let G-BanG \text{-Ban} be the category of Banach representations of GG (see “Analogy with the Finite Case”).

C c(G)=C(G)C_c(G) = C(G) is such a Banach representation. We may view \mathbb{R} as a Banach representation of GG as well, where gz=zgz = z for each zz \in \mathbb{R} and each gGg \in G. \mathbb{R} embeds into C(G)C(G) as constant functions. We may then consider the exact sequence

0C(G)C(G)/00 \rightarrow \mathbb{R} \rightarrow C(G) \rightarrow C(G)/ \mathbb{R} \rightarrow 0

A Haar integral on the GG-representation C(G)C(G) is equivalently a retract G:C(G)\int_G : C(G) \rightarrow \mathbb{R} for the injection C(G)\mathbb{R} \rightarrow C(G). In other words, it is a function G:C(G)\int_G : C(G) \rightarrow \mathbb{R} such that

G(f 1+f 2)= Gf 1+ Gf 2f 1,f 2C(G) \int_G (f_1 + f_2) = \int_G f_1 + \int_G f_2 \forall f_1, f_2 \in C(G)
Gaf=a GffC(G),a \int_G a f = a \int_G f \forall f \in C(G), a \in \mathbb{R}
Gf g= GffC(G),gG \int_G f^g = \int_G f \forall f \in C(G), g \in G
C 0:|| Gf||C G||f||[G,] Top \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}}

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

In some sense, we might wish to show that Ext G-Ban 1(C(G),)\text{Ext}^1_{G \text{-Ban}}(C(G), \mathbb{R}) vanishes; this would show that the sequence

0C(G)C(G)/00 \rightarrow \mathbb{R} \rightarrow C(G) \rightarrow C(G)/ \mathbb{R} \rightarrow 0

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

Proof: We show that \mathbb{R} is an injective object in G-BanG \text{-Ban}. Take an injection of Banach representations of GG, XYX \rightarrow Y. Let f:Xf : X \rightarrow \mathbb{R} be a map of Banach representations of GG. By the (usual) Hahn-Banach theorem, there exists a map g:Yg : Y \rightarrow \mathbb{R} in the category of Banach spaces and short maps extending ff, though it may lack GG-invariance.

Consider the subset of all extensions of ff to YY. Let SS be the collection of GG-invariant compact convex subsets of this set. SS contains the convex hull of GgG g, where gg is some chosen extension of ff to YY, so SS is nonempty. Using compactness and Zorn's Lemma, we may find a minimal element of SS in this collection, where SS is ordered where ABA \leq B when ABA \subset B. Call this element HH. HH must be a singleton. If HH contains a point which is not extremal then it contains the convex hull of the orbit of that point, which would be a proper GG-invariant compact convex subset of HH (see Krein Milman theorem). Therefore HH is a singleton, and its unique element is a GG-invariant functional extending ff.

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

0C(G)0 \rightarrow \mathbb{R} \rightarrow C(G)

giving a retract G:C(G)\int_G : C(G) \rightarrow \mathbb{R} for 0C(G)0 \rightarrow \mathbb{R} \rightarrow C(G) in G-BanG \text{-Ban}.

It follows that G\int_G has norm 11, and from this positivity follows immediately.

Extensive and Intensive Properties

In Lawvere’s thinking about extensive and intensive quantities,

  • C(G)C(G) is a space of intensive quantities on GG.

  • [C(G),] Ban[C(G), \mathbb{R}]_{\text{Ban}} is the space of extensive quantities on XX, where Ban\text{Ban} is the category of Banach spaces with bounded maps as maps. The Haar integral is the unique GG-invariant element of this space of norm 11.

  • the integration map is the canonical evaluation pairing

    G:C(G)×[C(G),] Ban. \int_G \;\colon\; C(G) \times [C(G), \mathbb{R}]_{\text{Ban}} \longrightarrow \mathbb{R} \,.

If we suggestively write Gfdϕ\int_G f d \phi for G(f,ϕ)=ϕ(f)\int_G (f, \phi) = \phi(f), then Gdϕ\int_G - d \phi becomes a way of writing ϕ\phi. In particular, choosing ϕ\phi to be the Haar measure, we can write ϕ\phi as Gdϕ\int_G - d \phi.

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

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

The left and right invariant measures are, respectively,

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

and so G is not unimodular.

Abelian groups are obviously unimodular; so are compact groups and discrete groups.

References

Last revised on April 8, 2021 at 02:23:49. See the history of this page for a list of all contributions to it.