# A characterization of L^p?

Could someone confirm or refute the following statement?

$$f \in L^p\left(X, \mu\right) \: \Leftrightarrow \: \int_X{\lvert fg \rvert d\mu < \infty\: \forall g \in L^q\left(X, \mu\right)$$

where $$1<p<\infty,\: \frac{1}{p}+\frac{1}{q}=1$$ and $$(X, \mu)$$ is a measurable space (of course, the $$(\Rightarrow)$$ is trivial by Holder inequality)