If $f, g \in L^p$, is it true that $\int | f g | = \int | f | \int | g |$? Let $f,g \in L^p(0, 1), \;\; 1 < p < \infty$. In this case, is it true that 
$$\underset{(0, 1)}{\int} | f(x) g(x) | dx = \underset{(0, 1)}{\int} | f(x) | dx \underset{(0, 1)}{\int} | g(x) | dx? $$
Why yes/ not?
Thank you!

Counterexample: Consider $(0,1)$ with Lebesgue measure and the characteristic functions $\chi_{(0,\frac{1}{2})}$ and $\chi_{(\frac{1}{2},1)}$.  Then $\chi_{(0,\frac{1}{2})}\chi_{(\frac{1}{2},1)} = 0$ but the product of their integrals is $\frac{1}{4}$.
