  Given a multi-index sequence $\mu_{\mathbf{k}}$, $\mathbf{k} = (k_1,..., k_n)
\in \mathbb{N}_0^n$, necessary and sufficient conditions are given for the
existence of a regular Borel polymeasure $\gamma$ on the unit interval $I=
[0,1]$ such that $\mu_{\mathbf{k}} = \int_{I^n} t_1^{k_1}\otimes ... \otimes
t_n^{k_n} \gamma$. This problem will be called the weak multilinear Hausdorff
problem of moments for $\mu_{\mathbf{k}}$. Comparison with classical results
will allow us to relate the weak multilinear Hausdorff problem with the
multivariate Hausdorff problem. A solution to the strong multilinear Hausdorff
problem of moments will be provided by exhibiting necessary and sufficient
conditions for the existence of a Radon measure $\mu$ on $[0,1]$ such that
$L_\mu(f_1,..., f_n) = \int_{I} f_1(t) ... f_n(t) \mu (dt)$ where $L_\mu$ is
the $n$-linear moment functional on the space of continuous functions on the
unit interval defined by the sequence $\mu_{\mathbf{k}}$. Finally the previous
results will be used to provide a characterization of a class of weakly
harmonizable stochastic processes with bimeasures supported on compact sets.
