I was working on an exercise from Tao's An Epsilon of Room to show the existence of Haar measure on an LCA group. For one of part of the problem we have ($G$ an LCA group, $C_c(G)^+$ denoting the subset of nonnegative not everywhere zero functions in $C_c(G)$):
Let $\mu$ be a Haar measure on $G$. Show for every $f \in C_c(G)^+$ and $\epsilon > 0$, there exists $g \in C_c(G)^+$ such that $\int_G f d\mu \ge (f :g) \int_G g d\mu - \epsilon$. (Hint: $f$ is uniformly continuous. Take $g$ to be an approximation to the identity.)
If you want to give the solution to the whole problem, that's all fine and dandy, but my actual questions are:
What does it mean to be uniformly continuous on an LCA group?
What is an approximation to the identity on an LCA group?
I'm unsure of these definitions basically because $G$ may not be a metric space.
So the definition I've ended up working with is:
$f$ is uniformly continuous on an LCA group $G$ if for each $\epsilon > 0$ we can pick a neighborhood $U$ of the identity such that $x- y \in U$ implies $|f(x) - f(y)|< \epsilon$.
As of right now I'm using Uryssohn lemma functions, possibly normalized in $L^1(G, \mu)$ norm, to be approximations to the identity.