The subsequent statement can be regarded as a follow-up to
- If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere
- Is $f$ non-negative a.e. if its primitive is non-decreasing?
Let $f:[a,b]\to\mathbb{R}$ be Lebesgue integrable. Furthemore, let $ g:[a,b]\ni x\mapsto\int_a^x f(t)\,\mathrm{d}t\in\mathbb{R} $ be convex. Then $f$ is non-decreasing almost everywhere.
Let $a\le x_0