  A scaling on some space is a measurable action of the group of positive real
numbers. A measure on a measurable space equipped with a scaling is said to be
$\alpha$-homogeneous for some nonzero real number $\alpha$ if the mass of any
measurable set scaled by any factor $t > 0$ is the multiple $t^{-\alpha}$ of
the set's original mass. It is shown rather generally that given an
$\alpha$-homogeneous measure on a measurable space there is a measurable
bijection between the space and the Cartesian product of a subset of the space
and the positive real numbers (that is, a "system of polar coordinates") such
that the push-forward of the $\alpha$-homogeneous measure by this bijection is
the product of a probability measure on the first component (that is, on the
"angular" component) and an $\alpha$-homogeneous measure on the positive
half-line (that is, on the "radial" component). This result is applied to the
intensity measures of Poisson processes that arise in
L\'evy-Khinchin-It\^o-like representations of infinitely divisible random
elements. It is established that if a strictly stable random element in a
convex cone admits a series representation as the sum of points of a Poisson
process, then it necessarily has a LePage representation as the sum of i.i.d.
random elements of the cone scaled by the successive points of an independent
unit intensity Poisson process on the positive half-line each raised to the
power $-\frac{1}{\alpha}$.
