  In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S.R. Moghadasi has
shown how the decomposition of the $N$-fold product of Lebesgue measure on
$\mathbb R^n$ implied by matrix polar decomposition can be used to derive the
Blaschke-Petkantschin decomposition of measure formula from integral geometry.
We use known formulas from random matrix theory to give a simplified derivation
of the decomposition of Lebesgue product measure implied by matrix polar
decomposition, applying too to the cases of complex and real quaternion
entries, and we give corresponding generalisations of the
Blaschke--Petkantschin formula. A number of applications to random matrix
theory and integral geometry are given, including to the calculation of the
moments of the volume content of the convex hull of $k \le N+1$ points in
$\mathbb R^N$, $\mathbb C^N$ or $\mathbb H^N$ with a Gaussian or uniform
distribution.
