Definition 28.3.1 (Modular Function).label Let $G$ be a locally compact group and $\mu$ be a left Haar measure on $G$, then
- (1)
For any $f, g \in C_{c}^{+}(G; \real) \setminus \bracs{0}$, $A, B \in \cb_{G}$ with $\mu(A), \mu(B) > 0$, and $y \in G$,
\[\Delta_{G}(y) = \frac{\int R_{y^{-1}}f d\mu}{\int f d\mu}= \frac{\int R_{y^{-1}}g d\mu}{\int g d\mu}= \frac{\mu(Ay)}{\mu(A)}= \frac{\mu(By)}{\mu(B)}> 0\] - (2)
For each $y \in G$ and $A \in \cb_{G}$, denote $\mu_{y}(A) = \mu(Ay)$, then $\mu_{y}(dx) = \Delta_{G}(y)\mu(dx)$.
- (3)
For any choice of $f \in C_{c}^{+}(G; \real) \setminus \bracs{0}$, the mapping $\Delta_{G}: G \to (0, \infty)$ defined by $y \mapsto \int R_{y^{-1}}f d\mu$ is a continuous group homomorphism.
- (4)
For each $A \in \cb_{G}$, let $\nu(A) = \mu(A^{-1})$, then $\nu(dx) = \Delta(x^{-1})\mu(dx)$.
The homomorphism $\Delta_{G}: G \to (0, \infty)$ is the modular function of $G$.
Proof, [Proposition 2.24, Proposition 2.31, Fol16]. (1), (2): For each $y \in G$, $\mu_{y}$ is also a left Haar measure. By [Haar’s Theorem]theorem:haar, there exists $\lambda > 0$ such that $\mu_{y} = \lambda \mu$. In which case,
(3): By Proposition 28.2.8, the mapping $y \mapsto \int R_{y} f d\mu$ is continuous. For any $x, y \in G$ and $A \in \cb_{G}$ with $\mu(A) > 0$,
(4): Let $f \in C_{c}^{+}(G)$ and $y \in G$, then
so $\Delta_{G}(x^{-1})\mu(dx)$ is a right Haar measure. By Haar’s Theorem, there exists $\lambda > 0$ such that $\nu(dx) = \lambda \Delta_{G}(x^{-1})\mu(dx)$.
If $\lambda \ne 1$, then there exists $U \in \cn_{G}(1)$ symmetric and compact such that $|\Delta_{G}(x^{-1}) - 1| \le 2^{-1}|\lambda - 1|$ on $U$. In which case, by symmetry, $\mu(U) = \nu(U)$, and
which contradicts the fact that $\lambda \ne 1$. Therefore $\lambda = 1$, and $\nu(dx) = \Delta_{G}(x^{-1})\mu(dx)$.$\square$
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