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**Erdos32212** Let $\displaystyle (X, \mathcal{B}, \mu)$ be a measure space. Use Hölder's inequality to prove that if $\displaystyle \frac{1}{p}+\frac{1}{q} = \frac{1}{r}$ and $\displaystyle r \geq 1$ then for every $\displaystyle f \in L^p(X, \mathcal{B}, \mu)$, $\displaystyle g \in L^q(X, \mathcal{B}, \mu)$, and $\displaystyle fg \in L^r(X, \mathcal{B}, \mu)$ with $\displaystyle || fg ||_r \leq || f ||_p \cdot || g ||_q$.

I don't see how to prove this right now. I know that this looks a lot like Young's inequality which follows from Hölder's inequality, so I was thinking to adopt a similar proof of that theorem. Right now, I just need a few hints on how to proceed. Thank you.