I'm trying to prove:
Let $X$ be a real random variable, $p, q \in (1,\infty)$, $\frac 1 p + \frac 1 q = 1$. If there is $C < \infty$ such that $|\mathbb E[XY]| \leq C ||Y||_q$ for any bounded random variable $Y$, then $X$ is in $\mathcal L^p$.
My idea is to use the fact that $\left(L^q(\mathbb P)\right)' \cong L^p(\mathbb P)$, and to show $F : L^q(\mathbb P) \to \mathbb R$ defined by $F(Y) = \mathbb E[XY]$ is continuous. For then, the isomorphism in particular is the isometry $\kappa(f) = \left( Y \mapsto \mathbb E[fY]\right)$, so we must have $f = X$. But I'm not sure if that conclusion is correct, nor am I sure how to prove $F$ is continuous over $L^q(\mathbb P)$; only over bounded functions in $L^q(\mathbb P)$. Any suggestions?