  Let $A$ be a CQG Hopf $*$-algebra, i.e. a Hopf $*$-algebra with a positive
invariant state. Given a unital right coideal $*$-subalgebra $B$ of $A$, we
provide conditions for the existence of a quasi-invariant integral on the
stabilizer coideal $B^{\perp}$ inside the dual discrete multiplier Hopf
$*$-algebra of $A$. Given such a quasi-invariant integral, we show how it can
be extended to a quasi-invariant integral on the Drinfeld double coideal. We
moreover show that the representation theory of the Drinfeld double coideal has
a monoidal structure. As an application, we determine the quasi-invariant
integral for the coideal $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$
constructed from the Podle\'{s} spheres.
