  By developing the hydrodynamic theory of spinor superfluids we calculate the
moment of inertia of a harmonically trapped Bose-Einstein condensate with
spin-orbit coupling. We show that the velocity field associated with the
rotation of the fluid exhibits diffused vorticity, in contrast to the
irrotational behavior characterizing a superfluid. Both Raman-induced and
Rashba spin-orbit couplings are considered. In the first case the moment of
inertia takes the rigid value at the transition between the plane wave and the
single minimum phase, while in the latter case the rigid value is achieved in
the limit of isotropic Rashba coupling. A procedure to generate the rigid
rotation of the fluid and to measure the moment of inertia is proposed. The
quenching of the quantum of circulation $h/m$, caused by Raman induced
spin-orbit coupling in a toroidal geometry, is also discussed.
