Lebesgue Integration Inequality

Question

This is a question from my measure class, and I don't really know how to approach this. Any hints would be helpful. Let $f,\ g$ be two increasing functions on $[0,1]$, measurable with respect to $m_1$. Prove that $$ \int_{[0,1]} fgdm_1 \geq \Big(\int_{[0,1]}fdm_1\Big)\cdot\Big(\int_{[0,1]}gdm_1\Big). $$

Answer 1

Just note that $(f(x)-f(y))(g(x)-g(y))\geq 0$ for all $x, y \in [0,1]\times [0,1]$; integrate this to get the answer.