The notation is indeed a bit non-standard (we would generally use intersections nowadays), but it's not completely ad hoc. If you think of P_n as the indicator

The notation is indeed a bit non-standard (we would generally use intersections nowadays), but it's not completely ad hoc. If you think of P_n as the indicator function of the polygon (i.e. P_n(x) = 1 if x is in the polygon and 0 otherwise), then it makes sense. (There is still a little bit of sloppiness from using P_n to denote the area of a polygon as well as the polygon itself.)