  We derive an explicit expression for the Haar integral on the quantized
algebra of regular functions C_q[K] on the compact real form K of an arbitrary
simply connected complex simple algebraic group G. This is done in terms of the
irreducible *-representations of the Hopf *-algebra C_q[K]. Quantum analogs of
the measures on the symplectic leaves of the standard Poisson structure on K
which are (almost) invariant under the dressing action of the dual Poisson
algebraic group K^* are also obtained. They are related to the notion of
quantum traces for representations of Hopf algebras. As an application we
define and compute explicitly quantum analogs of the Harish-Chandra c-functions
associated to the elements of the Weyl group of G.
