Definition 1

Let $g$ be a complex function on G and let Φ be a filter on G. We say that $g$ is slowly oscillating following Φ if, for every $\varepsilon >0$, there exists a set $M\in \Phi$ and a neighbourhood V of $e$ in G such that

$x\in M$ and $y\in V=\Rightarrow |g(xy)-g(x)|\leqslant \varepsilon$.