For all non negative measurable functions $f:B^n\rar[0,\infty]$:
$$
\int_{B^n} f(M_a(x))\,dx
=\int_{B^n} f(x)\Bigg(\frac{\sqrt{1-\Vert a\Vert^2}}{1-\la x,a\ra}\Bigg)^{n+1}\,dx
$$
Suppose $F:U\rar V$ is a diffeomorphism of open subsets $U,V$ of $\R^n$, $K$ a e.g. compact subset of $U$ and $g:V\rar[0,\infty]$ measurable. Then by the transformation (or substitution) formula of measure theory:
$$
\int_{F(K)} g(x)\,dx = \int_K g(F(x))|\det DF(x)|\,dx~.
$$
We just apply this formula to $K=B^n$, $g=f\circ M_a$ and $F=M_a^{-1}$ and use
proposition.