Let $M > 0$ and let $g: [0, 1] \rightarrow [0, \infty]$ be Borel measurable with $\int_{x}^{y} g(x) dx \leq M(y-x)$ for every $x < y \in [0, 1]$. Show $g \leq M$ a.e.
What I tried to do was tinker with $\int_{g > M} g(x) dx$, but I'm not sure how to relate it to the condition above with intervals. Is there a way I can approximate $\{g > M\}$ with intervals? I know I can approximate it with a Borel set (or even a closed set) but this doesn't seem to help! I can also say $g$ is continuous on a set that's close to $[0, 1]$ in measure.