Invariant integrals on coideals and their Drinfeld doubles

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.