Hölder's inequality of $L^{p}(X,\mu)$
$\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$
Log convexity of $L^{p}$ norm
$\left\Vert f \right\Vert_{r} \leq \left\Vert f \right\Vert_{p}^{1-\theta} \left\Vert f \right\Vert_{q}^{\theta}$ where $0<p,q,r\leq \infty, 0<\theta<1$ and $\frac{1-\theta}{p}+\frac{\theta}{q}=\frac{1}{r}$
How to prove the Hölder's inequality of $L^{p}(X,\mu)$ by the log convexity?
The hint is to manipulating $\mu$ and reduce to the case where $\mu(X)<\infty$,and then $f$ and $g$ are everywhere non vanishing simple functions.