  We describe all boundedly finite measures which are invariant by Cartesian
powers of an infinite measure preserving version of Chacon transformation. All
such ergodic measures are products of so-called diagonal measures, which are
measures generalizing in some way the measures supported on a graph. Unlike
what happens in the finite-measure case, this class of diagonal measures is not
reduced to measures supported on a graph arising from powers of the
transformation: it also contains some weird invariant measures, whose marginals
are singular with respect to the measure invariant by the transformation. We
derive from these results that the infinite Chacon transformation has trivial
centralizer, and has no nontrivial factor. At the end of the paper, we prove a
result of independent interest, providing sufficient conditions for an infinite
measure preserving dynamical system defined on a Cartesian product to decompose
into a direct product of two dynamical systems.
