I have functions continuous functions $f$, $g$ and $h$ on a bounded closed interval. I have an equation $|f - h + h - g| \leq |f - h| + |h - g|$. Can I integrate both sides of this equation over the interval?
Any reasonable integration theory states that $f \geq 0$ implies $\int_a^b f \geq 0$. Hence inequalities like $f \leq g$ can be integrated and yield $\int_a^b f \leq \int_a^b g$.