Folland , real analysis, theorem 6.14, dual of Lp spaces

Question

Somehow I feel Folland's proof of 6.14 is missing a small detail: in the construction of $f_n$, we used $||g_n||_q$, but how do we know $g_n$ is in $L^q$? Can someone please point out if I'm missing anything? Thanks!

Answer 1

Note that $\phi_n$ is simple function and $E_n$ is of finite measure. Then $g_n=\phi_n\chi_{E_n}\in L^p$ for any $p\in [1,\infty]$. Specifically, letting $$\phi_n=\sum_k a_k\chi_{A_k}$$ then $$\int|g_n|^p=\sum_{n}|a_k|^p\mu({E_n\cap A_k})\leq \max_j |a_j|^p\sum_k\mu(E_n\cap A_k)=\max_j|a_j|^p\mu(E_n)<\infty.$$