In this answer, I am confused by the following step. $$\int \hat{f} \hat{g}=\frac{1}{p} \int \hat{f}^p + \frac{1}{q} \int \hat{g}^p \iff \hat{f} \hat{g} = \frac{1}{p} \hat{f}^p + \frac{1}{q} \hat{g}^p \quad \text{a.e.}$$

The $\impliedby$ implication follows by integrating, but why does the $\implies$ implication hold?