What is an arithmetic mean filter?
In one picture: here, the mask is of size $m=3\times n=5$, filled with value $\frac{1}{3\times 5}$, the light gray shadow on the input image defines the neighborhood of the dark gray pixel: a set of pixels (all your $g(s,t)$) close to the center one (it moves with the pixel). Let us suppose the dark gray pixel has coordinates $x=5$ and $y=6$. The shadow defines $S_{5,6}$ having $s\in[5-(m-1),\dots,5+(m-1)]$ and $t\in[6-(n-1),\dots,6+(n-1)]$. $(s,t)\in S_{5,6} $ means for instance that the light gray pixel on the bottom left of the dark gray one, with coordinates $(6,5)$ belongs to the shadow.