Mathematics

In a measure space $(X,\mathcal{A},\mu)$ we consider two measurable functions $f,g:E\to\mathbb{R}$ for some $E\in\mathcal{A}$. We characterize the property of having equal $p$-norms when $p$ varies in an infinite set $P \in [1,+\infty)$. In a first theorem we consider the case of bounded functions when $P$ is unbounded with $\sum_{p\in P}(1/p)=+\infty\,$. The second theorem deals with the possibility of unbounded functions, when $P$ has a finite accumulation point in $[1,+\infty)$.