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Let $({\cal S}, \| \ \|) $ be a Banach space of jointly symmetric
$\alpha$-stable random variables and let ${\cal P} $ be the
$\delta$-ring of Borel sets of finite $\nu$ measure, where $\nu$
is a regular measure in the real line. In this paper we identify
every stable random measure $\Psi:{\cal P}\rightarrow ({\cal S},
\| \ \|), \Psi<< \nu$, by a vector measure $F :{ \cal
P}\rightarrow L^ {\alpha} (\mu)$. This leads to a method for
identifying spectral domain of a certain class of stable processes
including harmonizable processes.
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