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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 measureon $\Omega$. We prove a version of Beurling's invariant subspace theorem forthe space $L^{\beta}\left(\mu,H^{\alpha}\right) .$ Our proof uses the recentversion of Beurling's theorem on $H^{\alpha}\left(\mathbb{T}\right) $ proved bythe first author and measurable cross-section techniques. Our resultsignificantly extends a result of H. Rezaei, S. Talebzadeh, and D. Y. Shin.

Lebesgue and Hardy Spaces for Symmetric Norms II: A Vector-Valued Beurling Theorem