  We give a proof of the Thom-Sebastiani type theorem for holonomic filtered
$D$-modules satisfying certain good conditions (including Hodge modules) by
using algebraic partial microlocalization. By a well-known relation between
multiplier ideals and $V$-filtrations of Kashiwara and Malgrange, the argument
in the proof implies also a Thom-Sebastiani type theorem for multiplier ideals,
which cannot be deduced from a already known proof of the Thom-Sebastiani
theorem for mixed Hodge modules (since the latter gives only the information of
graded pieces of multiplier ideals). We also sketch a more elementary proof of
the Thom-Sebastiani type theorem for multiplier ideals (as communicated to us
by M.~Musta\c{t}\v{a}), which seems to be known to specialists, although it
does not seem to be stated explicitly in the literature.
