The space $L^{\infty -}$ and showing it is an algebra. Let 
$$\mathcal{A}=L^{\infty -}(\Omega,P):=\bigcap_{1\le p<\infty}L^p(\Omega,P)$$
I've already showed it is a vector space over $\mathbb{C}$. So, with the usual multiplication of (complex-valued) functions operation I want to prove that

$fg\in\mathcal{A}\quad\forall f,g\in\mathcal{A}\\\text{Which is the same as to prove that }\forall f,g\in\mathcal{A}\\
\int_{\Omega}|fg|^p\;dP<\infty,\quad 1\le p<\infty$

But got stucked. I only get that $fg\in L^1(\Omega,P)$ thanks to Hölder's inequality:
$$
\int_{\Omega}|fg|\;dP\le \Big(\int_{\Omega}|f|^p\,dP\Big)^{1/p}\Big( \int_{\Omega} |g|^q\;dP \Big)^{1/q},\quad\frac{1}{p}+\frac{1}{q}=1\quad 1\le p,q< \infty
$$
for $p=q=2$. From here I can't see how $fg\in L^p(\Omega,P)$ for $p\ge2$. Any help or hints would be appreciated.

Hint: $$|fg|^p = |f|^p |g|^p\le \frac{|f|^{2p} + |g|^{2p}}{2}$$
