Proposition 30.4.10 (Existence of Approximate Identity).label Let $G$ be a locally compact group and $\fB \subset \cn_{G}(1)$ be a fundamental system of neighbourhoods at $1$, directed under reverse inclusion. For each $U \in \fB$, let $\phi_{U} \in L^{+}(G)$ with $\ol{\bracsn{\phi_U \ne 0}}\subset U$ and $\int_{G} \phi_{U} = 1$, then:

1. (1)

   For each $p \in [1, \infty)$ and $f \in L^{p}(G; \complex)$, $\phi_{U} * f \to f$ in $L^{p}(G; \complex)$.

2. (2)

   For each left uniformly continuous $f \in C(G; \complex)$, $\phi_{U} * f \to f$ uniformly.

If in addition, $\phi_{U}(x^{-1}) = \phi_{U}(x)$ for all $x \in G$, then:

1. (3)

   For each $p \in [1, \infty)$ and $f \in L^{p}(G; \complex)$, $f * \phi_{U} \to f$ in $L^{p}(G; \complex)$.

2. (4)

   For each right uniformly continuous $f \in C(G; \complex)$, $f * \phi_{U} \to f$ uniformly.