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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. 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