On functions having coincident p-norms

In a measure space (X,A,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,{\mathcal {A}},\mu )$$\end{document}, we consider two measurable functions f,g:E→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g:E\rightarrow {\mathbb {R}}$$\end{document}, for some E∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\in {\mathcal {A}}$$\end{document}. We prove that the property of having equal p-norms when p varies in some infinite set P⊆[1,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\subseteq [1,+\infty )$$\end{document} is equivalent to the following condition: μ({x∈E:|f(x)|>α})=μ({x∈E:|g(x)|>α})for allα≥0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu (\{x\in E:|f(x)|>\alpha \})=\mu (\{x\in E:|g(x)|>\alpha \})\quad \text { for all } \alpha \ge 0. \end{aligned}$$\end{document}