Let $f : [0, 1] → \mathbb{R}$ be bounded and $f ≥ 0$. Prove that if the set $\{x\in[0,1]|f(x)\geq \lambda \}$ is finite for every $λ > 0$ then $f$ is integrable.
I'm having trouble understanding this criteria for integrability. I know that if a function is continuous on an interval then it is integrable on that interval and also that monotonicity implies integrability but I couldn't wrap my mind around this one. I assume that the set being finite implies monotonicity but I haven't really been able to show it, I'd appreciate help.