On functions having coincident p-norms

In a measure space (X, A, μ) , we consider two measurable functions f, g: E→ R, for some E∈ A. We prove that the property of having equal p-norms when p varies in some infinite set P⊆ [ 1 , + ∞) is equivalent to the following condition: μ(x∈E:|f(x)|>α)=μ(x∈E:|g(x)|>α)for allα≥0.