Definition 31.2.8.label Let $G$ be a locally compact group and assume that $G$ is equipped with a fixed left Haar measure $\mu: \cb_{G} \to [0, \infty]$, then

1. (1)

   For any normed vector space $E$ over $K \in \RC$, $L^{p}(G; E)$ denotes $L^{p}(G, \mu; E)$.

2. (2)

   For any $f \in L^{1}(G; E) \cup L^{+}(G)$, $\int_{G} f(x)dx$ denotes integration with respect to $\mu$.