  In this article, we give a definition for measured quantum groupoids. We want
to get objects with duality extending 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. To get a
satisfying duality in the general case, we assume the existence of an antipode
given by its polar decomposition. This theory is illustrated with many examples
among others inclusion of von Neumann algebras (M. Enock) and a sub family of
measured quantum groupoids with easier axiomatic.
