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.