  Suppose $\alpha$ is a rotationally symmetric norm on
$L^{\infty}\left(\mathbb{T}\right) $ and $\beta$ is a "nice" norm on
$L^{\infty}\left(\Omega,\mu \right) $ where $\mu$ is a $\sigma$-finite measure
on $\Omega$. We prove a version of Beurling's invariant subspace theorem for
the space $L^{\beta}\left(\mu,H^{\alpha}\right) .$ Our proof uses the recent
version of Beurling's theorem on $H^{\alpha}\left(\mathbb{T}\right) $ proved by
the first author and measurable cross-section techniques. Our result
significantly extends a result of H. Rezaei, S. Talebzadeh, and D. Y. Shin.
