Let $f$ be a nonnegative monotone function defined on the $[0,1]$. Prove that $$\mu_L(\{w\in I;f(w)>\alpha\})<\dfrac1\alpha\int_0^1fdx$$ where the measure is the Lebesgue measure, and the integral is the Riemann integral.
So let $f$ be monotone increasing (monotone decreasing should be similar). Suppose $f(w)=\alpha$ at $w=r$. So $\mu_L(\{w\in I;f(w)>\alpha\}) = 1-r$. Then partitioning $[0,1]$ into $[0,r],[r,1]$, we have $\int_0^1fdx$ is at least $\alpha(1-r)$. But still, it might be equal to $\alpha(1-r)$. How can we eliminate that possibility?