TITLE: Show that $g\in\mathcal{L}^q(\mu)$.
QUESTION [5 upvotes]: Let $(X,\mathcal{A},\mu$) be a finite measure space and $p,q\in(0,\infty)$ such that $1/p+1/q=1$. Let $g\in\mathcal{M}(\mathcal{A})$ measurable function such that 
$$\int |fg|d\mu\leq C\|f\|_p$$
for all $f\in\mathcal{L}^p(\mu)$ and some constant $C$.
Show that $g\in\mathcal{L}^q(\mu)$
I understand that Holder's Inequality does the reverse argument with $C=\|g\|_q$, but i don't know how to bend this into a proof.

REPLY [3 votes]: We use the relations $p>1;\ \  pq-p=q; \ \ p/(p-1)=q$, and the fact that 
$\mu(X)<\infty$, so that $\mu\{|g|>A\}<\infty$ too.
$$\int |g|^q1_{|g|\le N} d\mu =
 \int |g| \left[|g|^{q-1} 1_{|g|\le N}\right]d\mu
\\\le  C\left[
\int |g|^{(q-1)p} 1_{|g|\le N} d\mu
\right] ^{1/p}
 =  C\left[
\int |g|^{q} 1_{|g|\le N} d\mu
\right] ^{1/p} 
\\
\left[\int |g|^q1_{|g|\le N} d\mu \right] ^{1-1/p}
= \left[\int |g|^q1_{|g|\le N} d\mu \right] ^{1/q}
\le C
$$
As the bound does not depend on $N$, we conclude via the Fatou theorem:
$$
||g||_q \le C<\infty.
$$