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Throughout, let $(X,\mathfrak{M},\mu)$ be a measure space where $\mu$ is positive.The questionIf $f$ is of $L^p(\mu)$, which means $\lVert f \rVert_p=\left(\int_X |f|^p d\mu\right)^{1/p}<\infty$, or equivalently $\int_X |f|^p d\mu<\infty$, then we may say $|f|^p$ is of $L^1(\mu)$. In other words, we have a functional\begin{aligned}\lambda: L^p(\mu) &\to L^1(\mu) \\ f &\mapsto |f|^p.\end{aligned}This functional does not have to be one to one due to absolute value. But we hope this function...