Let $f:X \to C$ be measurable, $||f||_{\infty} > 0$, and let $\phi(p)= \int_X|f|^pd\mu$, and let $$E=\{p \in (0,\infty): \phi(p) < \infty\}$$ I would like to show that $\phi$ is continuous on $E$. I am really stuck on this. Any helps would be appreciated!
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If $p_n \to p$ then by continuity of $x \to a^x$ ,we have $|f|^{p_n} \to |f|^p$ pointwise a.s. when $|f| < \infty$. Now push DCT through. – Sarvesh Ravichandran Iyer Nov 05 '20 at 03:27