Let $G$ be a locally compact topological group. Let $\mu$ be left Haar measure on G. Suppose $f \in L^1(G)$. Then convolution with $f$ defines an operator from left uniformly continuous and bounded functions ($UCB(G)$) to itself. I am wondering if it is always true that given $\phi \in UCB(G)$, and $\varepsilon > 0$, there exists finitely many values $c_i \in \mathbb{C}$ and points $g_i \in G$ such that $|\sum c_i \ {}_{g_i} \phi - f * \phi|_\infty < \varepsilon$. Here $${}_{g_i}\phi (x) = \phi(g_i x)$$ and $$f * \phi (x) := \int_{G} f(y) \ {}_{y^{-1}}\phi(x) d \mu(y).$$
For context, in the book I'm reading (Property (T) by Bekka, De La Harpe, and Valette), in the section about amenability, they define amenability to mean the existence of a (finitely additive) invariant mean on $UCB(G)$, and then later claim that if $m$ is such a mean, then for any $f \in L^1(G)$ with $f \geq 0$ and $|f|_1 = 1$ and $\phi \in UCB(G)$, $m(f * \phi) = m (\phi)$. This result intuitively makes sense to me, but I am failing to see how to rigorously show this.