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\noindent\textbf{Corrigenda to ``Integration Theory: A Second Course''
by Martin V\"{a}th}
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This corrigendum is published so late, because I hoped for quite a while
that there will be a second edition of the book in which I could fix the
mistakes.
First of all, I want to excuse for falsely claiming after Theorem~7.1
that there is an inaccuracy in the corresponding proof of
J\"{u}rgen Elstrodt's book ``Ma\ss- und Integrationstheorie''.
This was a misunderstanding on \emph{my} side, and the alternative proof
of Elstrodt's book is completely correct.
A second mistake is in Proposition~2.8: It is in general true that
all measurable sets in a product $S_1\times\cdots\times S_n$ of Borel spaces
$S_i$ are Borel, but the converse holds in general only if all except at
most one of the spaces $S_i$ has a countable base of the topology.
(Thanks to J. Elstrodt for pointing out this mistake.)
Due to this mistake, one can claim in Theorem~5.3 only that the restriction
of the functions to $S\times S_0$ (and also to $S_0\times S$) are measurable
if $S_0\subseteq S$ has a countable base of its inherited topology.
The other assertions based on Theorem~5.3 in the book remain true,
nevertheless, but the proof of some results for convolutions is
more complicated if $S$ does not have a countable base of the topology:
Essentially, one must argue by approximating with functions
with compact support. On product spaces, one can show for such functions
a Fubini type theorem by means of the Stone-Weierstra\ss\ approximation
theorem. Details are already written in the second edition of the book,
but it is unclear whether this well ever appear.
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