Does $\Vert f \Vert_p = \sup_{\Vert g \Vert_q=1}\int fg d\mu$ fail if $f \notin L^p$?

Question

I know that for $p \in [1,\infty]$ if $X$ is $\sigma$-finite (for the $p=\infty$ case) we have $$ \Vert f \Vert_p = \sup_{\substack{g \in L^q\\\Vert g \Vert = 1}} \int_X fg d\mu. $$ I always see it stated assuming $f \in L^p$, eg on wikipedia, but I would like to apply it in a situation where this is possibly not the case. Does anyone know if it is still true in the case $\Vert f \Vert_p = \infty$?

Answer 1

If the space is $\sigma$-finite, it works. Indeed, let $\{A_n\}$ an increasing sequence of measurable sets of finite measure, and $g_n:=\chi_{A_n}\operatorname{sgn}(f_n)\chi_{\{|f|<n\}}|f|^{p-1}$. It's an increasing sequence, and by Fatou's lemma, $$+\infty\leqslant\liminf_{n\to+\infty}\int_X fg_n,$$ giving what we want.

When the space is not $\sigma$-finite, we can cheat, taking $X=\{a\}$ with the measure $\mu(\{a\})=+\infty$.