nLab Haar integral

Idea

For $G$ a locally compact Hausdorff topological group, with $C_c(G)$ denoting its algebra of compactly supported continuous functions from $G$ to $\mathbb{R}$, a Haar integral $\int_G$ is a continuous linear map $C_c(G) \rightarrow \mathbb{R}$ to the real numbers which is invariant under the obvious group action of $G$ on $C_c(G)$. 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 $G$-invariant Borel measure on $G$.

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

Definition

Let $G$ be a locally compact Hausdorff topological group. 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

$$\rho_K(f) \;=\; \sup_{x \in K} {|f(x)|} \,,$$

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

$$\int_G \,\colon\, C_c(G) \to \mathbb{R} \,.$$

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

$$\mu(B) \;=\; sup \Big\{ \textstyle{\int_G} f \,\colon\, supp(f) = K \subseteq B,\, \rho_K(f) = 1 \Big\} \,.$$

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

$$\int_G f \;\geq\; 0 \;\;\text{when}\; f \geq 0$$

$$\int_G f^g \;=\; \int_G f$$

for each $f \in C_c(G)$ and each $g \in G$, where $f^g \colon G \rightarrow \mathbb{R}$ sends $x$ to $f(g x)$.

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

$$\mu(g B) \;=\; \mu(B)$$

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

Properties

Existence and Uniqueness

Any locally compact Hausdorff topological group $G$ admits a Haar integral (and therefore Haar measure) that is unique up to scalar multiple. This result was first proven by André Weil. Proofs may be found in Alfsen 1963 (consructive) and Rubinstein-Salzedo 2004.

A particular argument for the case of a compact Hausdorff topological group was provided by Bader 2020, which is what we follow now:

Theorem

Let $G$ be a compact Hausdorff topological group. Write $C^0(G)$ for the Banach space of continuous $\mathbb{R}$-valued functions on $G$. There is a unique $G$-invariant $\mathbb{R}$-linear map of Banach-spaces $\int \colon C^0(G) \rightarrow \mathbb{R}$ such that $0 \leq \int f^2$ for all $f \in C^0(G)$.

Proof

Consider the set $S$ of subsets $C$ of $C^0(G)^*$ which satisfy

(1) ($G$-invariance) for all functionals $\phi$ in $C$, the continuous functional $\phi^g$ sending $f$ to $\phi(f^{g^{-1}})$ is contained in $C$.

(2) (compactness) The subtopology on $C$ is compact.

(3) (convexity) for elements $\phi_1, \ldots, \phi_n$ in $C$ and $a_1, ..., a_n \in [0,\infty)$ such that $\sum_{i = 1}^n a_i = 1$, $\sum_{i = 1}^n a_i \phi_i \in C$.

(4) (nonemptiness) there is an element of $C^0(G)^*$ contained in $C$.

Claim 1: $S$ is nonempty.

Recall that the Hahn-Banach theorem theorem for a Banach space with squared Minkowski seminorm $\text{sup}_{i \in I} \phi_i \overline{\phi_i}$ for continuous functionals $\{ \phi_i \}_{i \in I}$ provides the existence of a continuous functional $\phi : C^0(G) \rightarrow \mathbb{R}$. $\{ \phi \}$ satisfies (4) and (2) but possibly not (3) or (1).

The $\infty$-norm topology on $C^0(G)$ given by the squared Minkowski seminorm $\text{sup}_{g \in G} ev_g \overline{ev_g}$ is equivalently described as the compact open topology on continuous functions from $G$ to $\mathbb{R}$.

For any $f \in C^0(G)$, $\{ f \} \subset C^0(G)$ is closed because it is a single point in a Hausdorff space. Write $\mu_{g^{-1}} : G \rightarrow G$ for the continuous function sending $h$ to $hg^{-1}$. For a continuous function $f : G \rightarrow \mathbb{R}$, write $f^g : G \rightarrow \mathbb{R}$ for $f \circ \mu_{g^{-1}}$.

The composition

$$\text{ev} \circ \Big(1 \times \big(\mu \circ (1 \times \text{Inv}) \big) \Big) \;\colon\; [G,\mathbb{R}] \times G \times G \longrightarrow \mathbb{R}$$

is continuous, from which we get that

$$[G,\mathbb{R}] \times G \longrightarrow [G,\mathbb{R}]$$

is continuous, using the calculus of mates for exponentiable objects in topological spaces, along with the fact that a compact space is exponentiable. $[G,\mathbb{R}] \times G \rightarrow [G,\mathbb{R}]$ sending $(f,g)$ to the continuous function $f^g$ sending $h$ to $f(hg^{-1}) = f \circ \mu_{g^{-1}} (h)$ is continuous. It follows that image of $\{ f \} \times G$ in $[G,\mathbb{R}]$ is compact. This is the $G$-orbit $\{ f^g : g \in G \}$. $\{ f^g : g \in G \}$ satisfies (1) (2) and (4), but possibly not (3).

The Krein-Smulian theorem, applied to $G \cdot f$ which is compact in the Banach space $C^0(G)^*$, gives that the convex hull $\overline{G \cdot f}$ of $G \cdot f$ is compact in the weak topology on $C^0(G)^*$ (not to be confused with the weak${}^*$ topology on $C^0(G)^*$). We can conclude that $\overline{G \cdot f}$ is $G$-invariant, compact (relative to the weak topology), convex, and nonempty. Therefore $\overline{G \cdot f} \in S$.

Claim 2: given a chain $\{ C_n \}_{n \in \mathbb{N}}$ of elements $C_n \in S$ such that $C_{n+1} \subset C_n$, $\cap_{n \in \mathbb{N}} C_n \in S$.

(1) ($G$-invariance) if $\phi \in \cap_{n \in \mathbb{N}} C_n$, then $\phi \in C_n$ for each $n \in \mathbb{N}$, so $\phi^g \in C_n$ for each $n \in \mathbb{N}$, so $\phi^g \in \cap_{n \in \mathbb{N}} C_n$.

(2) (compactness) $\cap_{n \in \mathbb{N}} C_n$ is closed so

(3) (convexity) $\cap_{n \in \mathbb{N}} C_n$ is closed so

(4) (nonemptiness) $C_n$ is a decreasing chain of compact subsets of a Hausdorff topological space, from which we can conclude that it is nonempty.

Claim 3: Zorn's Lemma for the case of the partial order defined on $S$ by which $C_1 \leq C_2$ if and only if $C_2 \subseteq C_1$ is satisfied because of claim 2 and claim 3. This implies that we may find a minimal element of $S$ in this collection, where $S$ is ordered where $A \leq B$ when $A \subset B$.

Choose such an element $C$.

Claim 4: $C$ is a singleton.

The Krein-Milman theorem, applied to the compact, convex subset $C$ of $C^0(G)^*$ (again as a locally convex topological vector space under the weak topology), gives that $C$ is the convex hull of its extreme points.

The continuous group action of $G$ on $C$ sends the subspace $\text{Inext}(C)$ of inextremal points to itself, as the average $\sum_{i = 1}^n a_i f_i$ of $f_i$ with $a_i \neq 0$, $a_i \in (0,\infty)$, and $\sum_{i = 1}^n a_i$ remains an average with respect to the same weights $a_i \neq 0$, $a_i \in (0,\infty)$, with $\sum_{i = 1}^n a_i$:

$$(a_1 f_1 + a_2 f_2)^g = a_1 f_1^g + a_2 f_2^g$$

Since $G$ acts by homeomorphisms on $C$, $G$ must send the subspace of extremal points $\text{Ext}(C)$ of $C$ to itself. Using again that $G$ acts by homeomorphisms, we can conclude that $\phi$ is extremal if and only if $\phi^g$ is extremal.

For any $f \in \text{Inext}(C)$, one can construct an object in $S$ using the facts above (the compactness of any $G$-orbit and the resulting compactness of its convex hull $\tilde{C}$ within $C^0(X)^*$), and from that the inclusions $\{ f \} \subset \text{Inext}(C) \subsetneq C$ imply inclusions of the $G$-orbits $G \cdot \{ f \} \subset \text{Inext}(C) \subsetneq C$, which imply inclusions $\overline{G \cdot \{ f \}} \subset \text{Inext}(C) \subseteq C$, from which we can conclude that $C$ is not minimal, a contradiction!

Therefore $0 = \text{card}(\text{Inext}(C))$. This implies $1 = \text{card}(\text{Ext}(C))$, i.e. $C$ is a singleton, and its unique element a $G$-invariant functional extending $f$.

Extensive and Intensive Properties

In Lawvere’s thinking about extensive and intensive quantities,

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

• $[C(G), \mathbb{R}]_{\text{Ban}}$ is the space of extensive quantities on $X$, where $\text{Ban}$ is the category of Banach spaces with bounded maps as maps. The Haar integral is the unique $G$-invariant element of this space of norm $1$.

• the integration map is the canonical evaluation pairing

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

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

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

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

The left and right invariant measures are, respectively,

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

and so G is not unimodular.

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

References

Textbook account:

• Glen Bredon, Section 0.3 of: Introduction to compact transformation groups, Academic Press (1972) [ISBN 9780080873596, pdf]

Proofs:

• E. M. Alfsen: A Simplified Constructive Proof of the Existence and Uniqueness of Haar Measure, Mathematica Scandinavica 12 (1963) [doi:10.7146/math.scand.a-10675]

• Simon Rubinstein-Salzedo: On the existence and uniqueness of invariant measures on locally compact groups (2004) [pdf]

• Uri Bader, MO answer (2020) [MO:a/351405]

See also:

• Uffe Haagerup, Agata Przybyszewska: Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces [arXiv:math/0606794]

• Wikipedia: Haar measure