  In this article, part of the author's thesis, we propose a definition for
measured quantum groupoid. The aim is the construction of objects with duality
including both quantum groups and groupoids. We base ourselves on J. Kustermans
and S. Vaes' works about locally compact quantum groups that we generalize
thanks to formalism introduced by M. Enock and J.M. Vallin in the case of
inclusion of von Neumann algebras. From a structure of Hopf-bimodule with left
and right invariant operator-valued weights, we define a fundamental
pseudo-multiplicative unitary. We introduce the notion of quasi-invariant
weight on the basis and, then, we construct an antipode with polar
decomposition, a coinvolution, a scaling group, a modulus and a scaling
operator. This theory is illustrated with different examples. Duality of
measured quantum groupoids will be discussed in a forthcoming article.
