Let be $1\leq p<\infty$ and $g$ a measurable funtion defined on $E$. I have to prove that if $fg\in L^p$ for every $f\in L^p(E)$, then $g$ is essentialy bounded, that is $g\in L^\infty (E)$.
I approached to this problem trying to prove the equivalent formulation: if $g\notin L^\infty (E)$, there exists a funcion $f\in L^p (E)$ such that $fg\notin L^p (E)$. I defined $E_n =\{x\in E : |g(x)|
$\displaystyle f=\sum\limits_{n=1}^{\infty} f_n \chi_{Q_n\cap G_n}$
wehere $f_n$ are constant funtions. I am in troble showing that $f$ belongs to $L^p$ if I choose, for example,
$\displaystyle f_n= \frac{1}{m(Q_n\cap G_n)n^{p+1}}$
but in particular I'm totally struck proving that $||fg||_p=\infty$, that is, $fg\notin L^p{E}$.