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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 $\int_Ef\cdot gdm=\|f\|_p^p$.

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  • $\begingroup$ In fact, if $f \in L^p(E)$ then $$\|f\|_p = \sup_{g \in L^q(E)} \int_E fg \, dm.$$ $\endgroup$ – Umberto P. Dec 4 '18 at 20:20
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Hint: try $g=\mathrm{sign}(f)\cdot|f|^\alpha$ for some $\alpha\in\mathbb R$.

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

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