$f^{p_1}$ belongs to $L^p(E)$ and $g = \chi_{E}$ belongs to $L^q(E)$?

Question

I was reading a Corollary in Royden, which states:

In the proof,

The book mentioned by observation that $f^{p_1}$ belongs to $L^p(E)$ and $g = \chi_{E}$ belongs to $L^q(E)$. I am not sure how this fact can be observed?

Answer 1

Notice that: $$||f^{p_1}||^{p}_p = \int\limits_{E} \, |f^{p_1}|^p \, \mathrm{d} \mu = \int\limits_{E} \, |f|^{\frac{p_1 p_2}{p_1}} \, \mathrm{d} \mu = \int\limits_{E} \, |f|^{p_2} \, \mathrm{d} \mu = ||f||^{p}_{p_2} < \infty \Rightarrow f^{p_1} \in L^p(E)$$ Additionally: $$||\chi_{E}||^{q}_q = \int\limits_{E} \, |\chi_{E}|^q \, \mathrm{d} \mu = \int\limits_{E} \, \chi_{E} \, \mathrm{d} \mu = \mu(E) < \infty \Rightarrow \chi_E \in L^q(E)$$