  Let $\mathcal A$ be a free hyperplane arrangement. In 1989, Ziegler showed
that the restriction $\mathcal A''$ of $\mathcal A$ to any hyperplane endowed
with the natural multiplicity $\kappa$ is then a free multiarrangement
$(\mathcal A'',\kappa)$. The aim of this paper is to prove an analogue of
Ziegler's theorem for the stronger notion of inductive freeness: if $\mathcal
A$ is inductively free, then so is the multiarrangement $(\mathcal
A'',\kappa)$. In a related result we derive that if a deletion $\mathcal A'$ of
$\mathcal A$ is free and the corresponding restriction $\mathcal A''$ is
inductively free, then so is $(\mathcal A'',\kappa)$ -- irrespective of the
freeness of $\mathcal A$. In addition, we show counterparts of the latter kind
for additive and recursive freeness.
