An exercise in Rudin's RCA Would you please give me some help on the following problem?
Suppose $1 \leq p \leq \infty$, and $q$ is the exponent conjugate to $p$. Suppose $\mu$ is a positive $\sigma$-finite measure and $g$ is a measurable function such that $fg\in L^1(\mu)$ for every $f\in L^p(\mu)$. Prove that then $g\in L^q(\mu)$. 
If $p=\infty$. It is obvious that $g\in L^1(\mu)$, so it suffices to consider the case $1\leq p \lt \infty$. I first tried to think of a counterexample by constructing some $g\notin L^q(\mu)$ with $fg\notin L^1$. But since the $\sigma$-finiteness of $\mu$ is given, I think that the proof should rely on the duality of $L^p$ and $L^q$. I guess solving the problem amounts to show that the map $\Psi:L^p \to \mathbb{C}$ defined by $\Psi(f)=\int fg$ is bounded, but I am stuck here.
 A: I'll suggest a brief sketch: Start by assuming that $f \ge 0$.

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*Choose sets $E_k$ with $\mu(E_k) < \infty$, $E_k \subseteq E_{k + 1}$, and $X = \bigcup E_k$ by $\sigma$-finiteness. Take $G_n = \{x : |g(x)| < n\}$, and let $$g_n = \chi_{G_n \cap E_k}\cdot g$$ be an indicator function. Note $g_n \in L^q$.


*Define (bounded) linear functionals $\Lambda_{n, k}$ on $L^p$ by
$$\Lambda_{n, k}(f) = \int g_n f d\mu$$


*If these have a common bound, we're done. If not, use the Banach-Steinhaus theorem to find $f$ for which $$\sup_{n, k} |\Lambda_{n, k} f| = \infty$$ Use the monotone convergence theorem to conclude that $gf \notin L^1$, a contradiction.
A: First try the easier problem that if there exists $k > 0$ such that $$\|fg\|_1 \le k \|f\|_p$$ for all $f \in L^p(\mu)$, then $g \in L^q(\mu)$. This is where $\sigma$-finiteness comes in.
Next define sets $$E_k = \left\{ f \in L^p(\mu) : \|fg\|_1 \le k \|f\|_p \right\}.$$ Each $E_k$ is closed and moreover $$L^p(\mu) = \bigcup_{k=1}^\infty E_k.$$ Apply the Baire Category Theorem. 
