Proposition 27.2.7.label Let $G$ be a locally compact group and $\mu: \cb_{G} \to [0, \infty]$ be a left/right Haar measure, then:
- (1)
For each $U \subset G$ open with $U \ne \emptyset$, $\mu(U) > 0$.
- (2)
For each $f \in C_{c}^{+}(G) \setminus \bracs{0}$, $\int \phi d\mu > 0$.
Proof, for left Haar measures. Since $\mu \ne 0$, there exists $g \in C_{c}^{+}(G) \setminus \bracs{0}$ such that $\int g d\mu > 0$. By compactness of $\supp{g}$, there exists $\seqf{x_j}\subset G$ such that:
\[\supp{g}\subset \bigcup_{j = 1}^{n} x_{j}^{-1}U \quad g \le \sum_{j = 1}^{n} L_{x_j}f\]
Thus $0 < \int g d\mu \le n\mu(U)$ and $0 < \int g d\mu \le n \int f d\mu$.$\square$
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