Abstract
Suppose P≔(pj)j=1∞ is a sequence of distinct real numbers pj>0. We prove that the equalities ‖f‖p=‖g‖p,p∈P, imply μ({x∈E:|f(x)|<α})=μ({x∈E:|g(x)|<α}),α≥0, whenever 0<μ(E)<∞ and f,g∈L∞(E) if and only if ∑j=1∞pjpj2+1=∞.
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