# If $f$ is integrable, then for $p < \infty$, $\int f^p <\infty$?

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.

• 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 $L^p$. – mathworker21 Oct 14 '18 at 6:56
• What do you mean by "magnitude"? – Sihyun Kim Oct 14 '18 at 7:00
• $|f_n| = |\varphi_n^{q-1}|$ – mathworker21 Oct 14 '18 at 7:00