Invariant integrals on coideals and their Drinfeld doubles

Let $A$ be a CQG Hopf $\ast$-algebra, i.e. a Hopf $\ast$-algebra with a positive invariant state. Given a unital right coideal $\ast$-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 $\ast$-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 $\ast$-algebra $U_q(\mathfrak{s}\mathfrak{l}(2,\mathbb{R}))$ constructed from the Podle'{s} spheres.