In a measure space \((X,{\mathcal {A}},\mu )\), we consider two measurable functions \(f,g:E\rightarrow {\mathbb {R}}\), for some \(E\in {\mathcal {A}}\). We prove that the property of having equal p-norms when p varies in some infinite set \(P\subseteq [1,+\infty )\) is equivalent to the following condition:

$$\begin{aligned} \mu (\{x\in E:|f(x)|>\alpha \})=\mu (\{x\in E:|g(x)|>\alpha \})\quad \text { for all } \alpha \ge 0. \end{aligned}$$

中文翻译：

---

关于具有一致p-范数的函数

在度量空间\（（X，{\ mathcal {A}}，\ mu} \）中，我们考虑了两个可测量函数\（f，g：E \ rightarrow {\ mathbb {R}} \），对于某些\ （E \在{\ mathcal {A}} \中）。我们证明当p在某些无限集\（P \ subseteq [1，+ \ infty）\）中变化时具有相等的p-范数的性质等于以下条件：

$$ \ begin {aligned} \ mu（\ {x \ in E：| f（x）|> \ alpha \}）= \ mu（\ {x \ in E：| g（x）|> \ alpha \ }）\ quad \ text {对于所有} \ alpha \ ge0。\ end {aligned} $$