Is this corollary in fact false when $p=1$?

Question

I'm reading the book "Measure Theory" written by V.I. Bogachev.

In the page 264 we have the following corollary.

4.4.5. Corollary. Let $\mu$ be a nonnegative finite or $\sigma$-finite measure and let $p^{-1}+q^{-1}=1$, where $1<p<\infty$. Suppose that a measurable function $f$ is such that $fg\in L^1(\mu )$ for all $g\in L^q(\mu )$. Then $f\in L^p(\mu )$.

You can see the proof of this corollary in this link.

This corollary is a consequence of the following proposition.

4.4.4. Proposition. Let $\mu$ be a nonnegative finite and $\sigma$-finite measure on a space $X$. A set $\mathcal{F}$ is bounded in $L^p(\mu)$, where $p\in [1,\infty )$, precisely when $\sup _{f\in > \mathcal{F}}\int _Xfgd\mu <\infty $ for all $g\in L^{p/(p-1)}(\mu )$.

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I want to know why the corollary restrict $p\in \color{red}{(}1,\infty )$. I don't see any reason to exclude the case $p=1$ since the proposition that the corollary relies accepts $p=1$.

My question is: is that restriction a typo or in fact there's a reason to exclude the case $p=1$?

Thank you for your attention!

Answer 1

It’s trivial if $p=1$. Simply take $g=1$ in that case, and your hypothesis implies $fg=f\in L^1$

Answer 2

Consider what $q$ would be in $p^{-1}+q^{-1}=1$.