  Let $\beta \equiv \{ \beta_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d}$ be
a $d$-dimensional multisequence. Curto and Fialkow, have shown that if the
infinite moment matrix $M(\beta)$ is finite-rank positive semidefinite, then
$\beta$ has a unique representing measure, which is $rank M(\beta)$-atomic.
Further, let $\beta^{(2n)} \equiv \{ \beta_\mathbf{i} \}_{\mathbf{i} \in
\mathbb{Z}_+^d, \mid \mathbf{i} \mid \leq 2n}$ be a given truncated
multisequence, with associated moment matrix $M(n)$ and $rank M(n)=r$, then
$\beta^{(2n)}$ has an $r$-atomic representing measure $\mu$ supported in the
semi-algebraic set $K=\{ (t_1, \ldots, t_d) \in \mathbb{R}^d : q_j(t_1, \ldots,
t_d) \geq 0, 1\leq j\leq m \}$, where $q_j \in \mathbb{R}[t_1, \ldots, t_d]$,
if $M(n)$ admits a positive rank-preserving extension $M(n+1)$ and the
localizing matrices $M_{q_j}(n +[\frac{\deg q_j +1}{2}])$ are positive
semidefinite; moreover, $\mu$ has precisely $rank M(n) - rank M_{q_j}(n
+[\frac{\deg q_j +1}{2}])$ atoms in $\mathcal{Z}(q_j) \equiv \{ t\in
\mathbb{R}^d: q_j(t)=0 \}$. In this paper, we show that every truncated moment
sequence $\beta^{(2n)}$ is a subsequence of an infinite recursively generated
multisequence, we investigate such sequences to give an alternative proof of
Curto-Fialkow's results and also to obtain a new interesting results.
