Proposition 1

Let $\mathbf{f}$ be a regulated vector function, $g$ a regulated function $\geqslant 0$ on an interval $[a, +\infty[$, such that $\int_a^{+\infty} g(t)\,dt > 0$. The relation $\mathbf{f} \preccurlyeq g$ as $x$ tends to $+\infty$ implies that $\int_a^x \mathbf{f}(t)\,dt \preccurlyeq \int_a^x g(t)\,dt$. If the integral $\int_a^{+\infty} g(t)\,dt$ converges then the integral $\int_a^{+\infty} \mathbf{f}(t)\,dt$ converges absolutely.

Indeed, by hypothesis there exist a $b \geqslant a$ and a number $c' > 0$ such that

whence since, moreover, one may suppose $b$ so large that $\int_a^b g(t)\,dt > 0$, there exists a $c'' > 0$ such that $\left\| \int_a^b \mathbf{f}(t)\,dt \right\| \leqslant c'' \int_a^b g(t)\,dt$; on putting $c = \max(c', c'')$ one thus has for every $x \geqslant b$, whence the proposition.