A question for a measurable function $g$ on a finite measure space such that $fgin L^p$ for all $fin L^p$

Question

Let $mu$ be a finite positive measure on $X$ and let $1leq p<infty$. Suppose that $g:Xto Bbb R$ satisfies $fgin L^p$ whenever $fin L^p(mu)$. I want to show that $||g||_infty =sup {||fg||_p:fin L^p(mu)$ such that $||f||_p=1}<infty$.
The inequality $||g||_infty geq sup {||fg||_p:fin L^p(mu)$ such that $||f||_p=1}$ is obvious. So it is left to show the converse inequality and that the value is finite, but I have no idea here. Any hints?

Kavi Rama Murthy · Answer

Hint: If $frac 1{mu (E)} int_E |g| dmuleq M$ for every set $E$ of positive  measure then $|g| leq M$ a.e. so $|g|_{infty} leq M$.
Now let $M=sup {|fg|_p: |f|_p=1}$. Take $f=frac 1 {mu (E)^{1/p}} chi_E$. Then $|f|_p=1$.  By Holder's inequality we have
$frac 1{mu (E)} int_E |g| dmuleq frac 1{mu (E)} |f(mu (E)^{1/p}) g|_p |I_E|^{q}$ where $frac 1 p+frac 1 q=1$. The desired equality now follows.
To show that $|g|_{infty} <infty$ consider $f =sum a_nchi_{n leq |g| <n+1}$ where $a_n$ are positive numbers. Let $c_n=mu {x: n leq |g(x)| <n+1}$. The hypothesis implies that $sum a_n^{p} n^{p}c_n <infty$ whenever $sum a_n^{p} c_n <infty$. I leave it to you to show that this can happen only when $c_n=0$ for $n$ sufficiently large. Of course this would then show that $g$ is essentially bounded.

A question for a measurable function $g$ on a finite measure space such that $fgin L^p$ for all $fin L^p$

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