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We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres $\\mathbb{S}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2(\\mathbb{S}^d)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p(\\mathbb{S}^d), p<+\\infty$ spaces. 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