  The Haar functional on the quantum $SU(2)$ group is the analogue of invariant
integration on the group $SU(2)$. If restricted to a subalgebra generated by a
self-adjoint element the Haar functional can be expressed as an integral with a
continuous measure or with a discrete measure or by a combination of both.
These results by Woronowicz and Koornwinder have been proved by using the
corepresentation theory of the quantum $SU(2)$ group and Schur's orthogonality
relations for matrix elements of irreducible unitary corepresentations. These
results are proved here by using a spectral analysis of the generator of the
subalgebra. The spectral measures can be described in terms of the
orthogonality measures of orthogonal polynomials by using the theory of Jacobi
matrices.
