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/Part 4: Measure Theory and Integration/Chapter 27: Integration on Locally Compact Groups

27.2 Haar Measures

Definition 27.2.1 (Haar Measure).label Let $G$ be a locally compact group and $\mu: \cb_{G} \to [0, \infty]$ be a non-zero Radon measure, then $\mu$ is a left Haar measure if

  1. (LH)

    For each $g \in G$ and $A \in \cb_{G}$, $\mu(gA) = \mu(A)$.

and a right Haar measure if

  1. (RH)

    For each $g \in G$ and $A \in \cb_{G}$, $\mu(Ag) = \mu(A)$.

Lemma 27.2.2.label Let $G$ be a locally compact group, then there exists an open and closed subgroup $H$ that is $\sigma$-compact.

Proof. Let $K \in \cn_{G}(1)$ and $K^{(1)}= K$. For each $n \in \natp$, let $K^{(n+1)}= KK^{(n)}$, then $K^{(n+1)}$ is compact by Proposition 5.16.2 with $K^{(n+1)}\in \cn_{G}(K^{(n)})$. Let $H = \bigcup_{n \in \natp}K^{(n)}$, then $H$ is a subgroup of $G$, which is open by Lemma 5.4.3. Since $H$ admits an exhaustion by compact sets, it is $\sigma$-compact.

Finally, since

\[G \setminus H = G \setminus \bigcup_{n \in \natp}K^{(n)}= \bigcap_{n \in \natp}G \setminus K^{(n)}\]

and $K^{(n)}$ is closed for each $n \in \natp$ by Proposition 5.16.3, $G \setminus H$ is closed, and hence $H$ is open.$\square$

Definition 27.2.3 (Covering Ratio).label Let $G$ be a locally compact group and $f, g \in C_{c}^{+}(G)$, then

\[(f: g) = \inf\bracs{\sum_{j = 1}^n c_j \bigg | \seqf{c_j} \subset [0, \infty), \seqf{x_j} \subset G, f \le \sum_{j = 1}^n c_j L_{x_j}g}\]

is the covering ratio of $f$ by $g$.

Proposition 27.2.4.label Let $G$ be a locally compact group and $f, h, g \in C_{c}^{+}(G)$, then:

  1. (1)

    If $g \ne 0$, then $(f: g) < \infty$.

  2. (2)

    $(f, h: g) \le (h: g) + (h: g)$.

  3. (3)

    For each $\lambda \ge 0$, $(\lambda f: g) = \lambda(f: g)$.

  4. (4)

    If $f \le h$, then $(f: g) \le (h: g)$.

  5. (5)

    $(f: g) \le (f: h)(h: g)$.

  6. (6)

    $(f: g) \ge \norm{f}_{u}/\norm{g}_{h}$.

  7. (7)

    For each $x \in G$, $(L_{x}f: g) = (f: g)$.

Lemma 27.2.5.label Let $G$ be a locally compact group, $f, f'\in C_{c}^{+}(G)$, and $\eps > 0$, then there exists $V \in \cn_{G}(1)$ such that for any $g \in C_{c}^{+}(V)$ with $g \ne 0$,

\[(f: g) + (f': g) \le (f + f': g) + \eps\]

Proof, [Lemma 2.18, Fol16]. By Urysohn’s Lemma, there exists $\eta \in C_{c}^{+}(G; [0, 1])$ such that $\eta|_{\supp{f} \cup \supp{f'}}= 1$.

Let $\delta > 0$, and define

\[H = f + f' + \delta \eta \quad h = \frac{f}{H}\quad h' = \frac{f'}{H}\]

By Proposition 27.1.2, there exists $V \in \cn_{G}(1)$ such that for any $x, y \in G$ with $x^{-1}y \in V$,

\[|h(x) - h(y)|, |h'(x) - h'(y)| < \delta\]

Let $g \in C_{c}^{+}(V)$, $\seqf{c_j}\subset [0, \infty)$, and $\seqf{x_j}\subset G$ such that $H \le \sum_{j = 1}^{n} c_{j} L_{x_j}\phi$, then for each $x \in G$,

\begin{align*}f(x)&= H(x)h(x) \le \sum_{j = 1}^{n} c_{j} L_{x_j}g(x)h(x) = \sum_{j = 1}^{n} c_{j}g(x_{j}^{-1}x)h(x) \\&\le \sum_{j = 1}^{n} c_{j}[h(x_{j}) + \delta] \cdot L_{x_j}g(x)\end{align*}

Likewise,

\[f'(x) \le \sum_{j = 1}^{n} c_{j}[h'(x_{j}) + \delta] \cdot L_{x_j}g(x)\]

As $h + h' \le 1$,

\[(f: g) + (f': g) \le \sum_{j = 1}^{n} c_{j}[h(x_{j}) + h'(x_{j}) + 2\delta]\]

Since the above holds for all such $\seqf{c_j}\subset [0, \infty)$ and $\seqf{x_j}\subset G$,

\begin{align*}(f: g) + (f': g)&\le (1 + 2\delta)(H: g) \\&\le (1 + 2\delta)[(f + f': g) + \delta(\eta: g)]\end{align*}

$\square$

Theorem 27.2.6 (Haar).label Let $G$ be a locally compact group, then:

  1. (1)

    There exists a left/right Haar measure on $G$.

  2. (2)

    For any two left/right Haar measures $\mu$ and $\nu$ on $G$, there exists $\lambda > 0$ such that $\mu = \lambda \nu$.

Proof, [Theorem 2.10, 2.20, Fol16]. (1): Fix $h \in C_{c}^{+}(G)$ with $h \ne 0$. For each $g \in C_{c}^{+}(G)$ with $g \ne 0$, let

\[I_{g}: C_{c}^{+}(G) \to [0, \infty) \quad f \mapsto \frac{(f: g)}{(h: g)}\]

then by (5) of Proposition 27.2.4, for each $f \in C_{c}^{+}(G)$ with $f \ne 0$,

\[\frac{1}{(h: f)}= \frac{(f: g)}{(h: f)(f: g)}\le I_{g}(f) \le \frac{(f: h)(h: g)}{(h: g)}= (f: h)\]

Thus $\mathcal{I}(f) = \bracs{I_g(f)|g \in C_c^+(G) \setminus \bracs{0}}$ is precompact for each $f \in C_{c}^{+}(G)$.

For each $V \in \cn_{G}(1)$, let $E_{V} = \bracs{I_g|g \in C_c^+(V) \setminus \bracs{0}}$, then $\fF = \bracs{E_V|V \in \cn_G(1)}$ is a filter on the product space $\prod_{f \in C_c^+(G)}\ol{\mathcal{I}(f)}$. By Tychonoff’s Theorem, there exists $\bigcap_{V \in \cn_G(1)}\ol{E_V}\ne \emptyset$.

Let $I \in \bigcap_{V \in \cn_G(1)}\ol{E_V}$, then by continuity,

  1. (i)

    For each $f \in C_{c}^{+}(G) \setminus \bracs{0}$, $I(f) \in [(h: f)^{-1}, (f: h)]$.

  2. (ii)

    For every $\lambda \ge 0$ and $f \in C_{c}^{+}(G)$, $I(\lambda f) = \lambda I(f)$.

  3. (iii)

    For any $x \in G$, $I(L_{x}f) = I(f)$.

  4. (iv)

    For each $f, f' \in C_{c}^{+}(G)$, $I(f + g) \le I(f) + I(g)$.

Let $f, f' \in C_{c}^{+}(G)$ and $\eps > 0$. By Lemma 27.2.5, there exists $V \in \cn_{G}(1)$ such that for each $g \in E_{V}$,

  1. (a)

    $|I_{g}(f) - I(f)|, |I_{g}(f') - I(f')| < \eps$.

  2. (b)

    $I_{g}(f) + I_{g}(f') \le I_{g}(f + f') + \eps$.

In which case, $I(f) + I(f') \le I(f + f') + 3\eps$. Since this holds for all $\eps > 0$,

  1. (v)

    For each $f, f' \in C_{c}^{+}(G)$, $I(f + g) \ge I(f) + I(g)$.

Using Lemma 16.1.13, $I$ extends to a positive linear functional on $C_{c}(G; \real)$, with $I(f) > 0$ for all $f \in C_{c}^{+}(G) \setminus \bracs{0}$, and

  1. (LH)

    For each $f \in C_{c}(G)$ and $x \in G$, $I(L_{x}f) = I(f)$.

By (i) and the Riesz Representation Theorem, there exists a unique non-zero Radon measure $\mu: \cb_{G} \to [0, \infty]$ such that for each $f \in C_{c}^{+}(G)$, $I(f) = \int_{G} f d\mu$. Finally, by density of $C_{c}(\mu; \real)$ in $L^{1}(\mu; \real)$ and (LH), $\mu$ is a left Haar measure.$\square$

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