Functions with identical \(L_p\) norms (Q783724)

Let \((E,\mathcal{A},\mu)\) be a measure space with \(0<\mu(E)<\infty\) and let \[ {\|f\|}_p:=\Bigl(\int_E|f(x)|^p\,d\mu(x)\Bigr)^{1/p} \] and \[ {\|f\|}_{\infty}:=\inf\bigl\{\alpha\in{\mathbb R}:\mu(\{x\in{E}:|f(x)|>\alpha\})=0\bigr\}. \] Let \(f,g\in{L}_{\infty}(E)\) and suppose that \(P:=(p_j)_{j=1}^{\infty}\) is a sequence of distinct real numbers \(p_j>0\). In this paper, the author proves that the equalities \[ {\|f\|}_p={\|g\|}_p, \qquad p\in{P}, \] imply \[ \mu(\{x\in{E}:|f(x)|<\alpha\})=\mu(\{x\in{E}:|g(x)|<\alpha\}), \qquad \alpha\geq0, \] if and only if \(\sum_{j=1}^{\infty}\frac{p_j}{p_j^2+1}=\infty\).