  We are going to widen the scope of the previously defined Hausdorff-integral
in two ways. First, in the sense, that we develop the theory of the integral on
some naturally generalized measure spaces. Second, we extend it to functions
taking values in $[0,+\infty)\times[0,+\infty)$. In all our intentions, we
follow the same attitude that we had in our previous investigation, i.e. we
work in the realm of Hausdorff dimension and measure.
