  Lebesgue integration is a well-known mathematical tool, used for instance in
probability theory, real analysis, and numerical mathematics. Thus its
formalization in a proof assistant is to be designed to fit different goals and
projects. Once Lebesgue integral is formally defined and the first lemmas are
proved, the question of the convenience of the formalization naturally arises.
To check it, a useful extension is the Tonelli theorem, stating that the
(double) integral of a nonnegative measurable function of two variables can be
computed by iterated integrals, and allowing to switch the order of
integration. Therefore, we need to define and prove results on product spaces,
hoping that they can easily derive from the existing ones on a single space.
This article describes the formal definition and proof in Coq of product
$\sigma$-algebras, product measures and their uniqueness, the construction of
iterated integrals, up to the Tonelli theorem. We also advertise the
\emph{Lebesgue induction principle} provided by an inductive type for
{\nonnegative} measurable functions.
