Is the following proof to Hölders inequality correct ? Hölders inequality is $\int |fg|dx \le||f||_p||g||_q$ 
Define $F(x)= \frac{f(x)}{(g(x))^{q/p}}$ and $\nu dx =g(x)^q$ 
and $\Phi(t)= |t|^p$,  $p\in (1,\infty)$ 
Now lets find $\Phi(\int F(x) d\nu)= \frac{(\int fg)^p}{||g||_q^{qp}}$ 
and similarly find $\int \Phi (F(x))$ and apply jensens inequality . 
My doubt is in the setting of $F(x)$ and $\nu dx $
 Can anyone help me to make this proof correct . 
Thanks 
 A: We assume WLOG that $f$ and $g$ are non-negative. We should take $F(x):=\frac{f(x)}{(g(x))^{q/p}}\chi_{g\neq 0}$ and we have to make the change of measure, if $\mu$ is the initial one $\nu=\frac{g(x)^q}{\lVert g\rVert^q_q}\mu$, that is, if $h$ is integrable 
$$\int h(x)\nu(dx)=\int h(x)g(x)^qd\mu.$$
We can apply Jensen's inequality since $\nu$ is a probability measure. We have 
\begin{align}
\Phi(\int F(x)d\nu(x))&=\left|\int F(x)d\nu(x)\right|^p\\
&=\left|\int F(x)\frac{g(x)^q}{\lVert g\rVert^q_q}d\mu(x)\right|^p\\
&=\frac 1{\lVert g\rVert^{qp}_q}\left|\int\frac{f(x)}{(g(x))^{q/p}}\frac{g(x)^q}{\lVert g\rVert^q_q}\chi_{g\neq 0}d\mu(x)\right|^p\\
&=\frac 1{\lVert g\rVert^{2qp}_q}\left|\int fgd\mu(x)\right|^p,
\end{align}
since $q-q/p=q/q=1$. We also have 
\begin{align}
\int\Phi(F(x))d\nu(x)&=\int|F(x)^p|d\nu\\
&=\int |F(x)|^p\frac{g(x)^q}{\lVert g\rVert^q_q}d\mu(x)\\
&=\int \left|\frac{f(x)}{(g(x))^{q/p}}\chi_{g\neq 0}\right|^p\frac{g(x)^q}{\lVert g\rVert^q_q}d\mu(x)\\
&=\frac 1{\lVert g\rVert_q^q}\int |f(x)|^pd\mu(x).
\end{align}
