Bi-$M$-invariant measure on a Riemannian symmetric space

Question

Let $G$ be a noncompact connected semi simple Lie group. Let $K$ be a maximal compact subgroup and $G=K\overline{A_{+}}K$ be a Cartan decomposition of $G.$ Let $M=Z_{K}(\mathfrak{a})$. Then how to describe the double coset space. Do we have simple description of the integral $\int_{K}f(k)dk$ for a bi-$M$-invariant integrable function $f$?