Let $G$ be a locally compact second countable topological group (Lie group if it helps). A function $f : G \to \mathbb{C}$ is left uniformly continuous if for all $\epsilon > 0$ there exists an open neighborhood $U$ of the identity such that $\sup_{g \in G} |f(ug) - f(g)| < \epsilon$ for all $u \in U$.
The space $LUC(G)$ of such functions is a sup-norm closed subalgebra of $C(G)$. My question is whether it is dense in $L^1(G,Haar)$, i.e. is it true that for any $F \in L^{1}(G,Haar)$ and $\epsilon > 0$ there exists $f \in LUC(G)$ such that $\|F - f\|_{1} < \epsilon$, and if so, if the same holds with the further restrictions that $f$ be compactly supported and bounded.
