An application of Holder's Inequality

Question

Suppose $1\leq p,q\leq \infty$ and $1/p+1/q=1$. Let $f\in\mathcal{L}^p(E)$. Show that $f=0$ a.e. if and only if \begin{align*} \int_E f\cdot gdm=0 \end{align*} for all $g\in \mathcal{L}^q(E)$. Hint: Choose a smart $g$ so that $\int_Ef\cdot gdm=\|f\|_p^p$.

Answer 1

Hint: try $g=\mathrm{sign}(f)\cdot|f|^\alpha$ for some $\alpha\in\mathbb R$.

Second hint: $\alpha=p/q$.