  The $\kappa$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$ and
$\mathscr{M}^1$ are defined by a neighbourhood basis consisting of polars of
absolutely convex and compact subsets of their (pre-)dual spaces. In many cases
it is more convenient to work with a description of the topology by means of a
family of semi-norms defined by multiplication and/or convolution with
functions and by classical norms. We give such families of semi-norms
generating the $\kappa$-topologies on the above spaces of functions and
measures defined by integrability properties. In addition, we present a
sequence-space representation of the spaces $\mathscr{D}_{L^p}$ equipped with
the $\kappa$-topology, which complements a result of J.~Bonet and M.~Maestre.
As a byproduct, we give a characterisation of the compact subsets of the spaces
$\mathscr{D}'_{L^p}$, $L^p$ and $\mathscr{M}^1$.
