  This paper is concerned with polynomially generated multiplier invariant
subspaces of the weighted Bergman space $A_{\boldsymbol{\beta}}^2$ in
infinitely many variables. We completely classify these invariant subspaces
under the unitary equivalence. Our results not only cover cases of both the
Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ and the Bergman space
$A^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables, but also apply
in finite-variable setting.
