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To be $f_n \in L^p$, it should satisfy $\int_E |f_n|^p < \infty$. I understand that $\varphi_n$ is integrable since it is dominated by an integrable function $g$. But, does this imply that for $q <\infty$, $\int_E |f_n|^p = \int_E |\varphi_n|^{q-1} <\infty$ ?

In general, if some exponents, $p$, are finite, does this imply that $\int_E f^p < \infty$ for an integrable $f$?

I appreciate if you give some help.

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  • $\begingroup$ you're generalizing too much. $f_n$ has the same magnitude as $\varphi_n^{q-1}$, so $f_n \in L^p$ iff $\varphi_n^{q-1} \in L^p$, but $\varphi_n^{q-1}$ is simple, so of course it's in $L^p$. $\endgroup$ – mathworker21 Oct 14 '18 at 6:56
  • $\begingroup$ What do you mean by "magnitude"? $\endgroup$ – Sihyun Kim Oct 14 '18 at 7:00
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    $\begingroup$ $|f_n| = |\varphi_n^{q-1}|$ $\endgroup$ – mathworker21 Oct 14 '18 at 7:00

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