closure in $G$ is $\overline{A_{+}}=\exp \overline{\mathfrak{a}
_{+}}$. We have the Cartan decomposition $$G=K\overline{A_{+}}K=K\exp \overline{\mathfrak{a}_{+}}K.  \label{kak}$$ Then, each element $x \in G $ is written uniquely as $x =k_1(\exp H)k_2$. We set $$\label{normG}
|x|=|H|, \; H \in \overline{\mathfrak{a}_{+}},$$ the norm on $G$ [@ANO p.2]. Denote by $x_0 = eK$ a base point of $X$. If $x,y\in X$, there are isometries $g,h \in G $ such that $x = gx_0$ and $y
= hx_0$. Because of the Cartan decomposition (\[kak\]), there are $k,
k^{\prime}\in K$ and a unique $H \in \overline{\mathfrak{a}_{+}}$ with $%
g^{-1}h=k \exp H k^{\prime}$ . It follows that $$\ d(x, y)=|H|,$$ where $d(x,y)$ is the geodesic distance on $X$ [@Weber].

Normalize the Haar measure $dk$ of $K$ such that $\int_{K}dk=1$. Then, from the Cartan decomposition, it follows that $$\label{haarmeas}
\int_{G}f(g)dg=\int_{K}dk_{1}\int_{{\mathfrak{a}_{+}}}{\delta (H)}%
dH\int_{K}f(k_{1}\exp (H)k_{2})dk_{2},$$ where the modular function $\delta (H)$ satisfies the estimate $$\label{modular}
\delta(H)\leq ce^{2\rho (H)} .$$ We identify functions on $X = G/K$ with functions on $G$ which are $K$-invariant on the right, and hence bi-$K$-invariant functions on $G$ are identified with functions on $X$, which are $K$-invariant on the left. Note that if $f$ is $K$-bi-invariant, then by (\[haarmeas\]), $$\int_{G}f\left( g\right) dg=\int_{X}f\left( x\right) dx=c\int_{\mathfrak{a}%
_+}f(\exp H)\delta (H)dH.  