Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval

Let $n$ be a natural number, let $a\le b$ be real numbers, and let $g=(g^1,\dots,g^n):[a,b]\to\mathbb{R}^n$ be a map into Euclidean space whose components are bounded and measurable with respect to the trace Borel $\sigma$-algebra on $[a,b]$ and the Borel $\sigma$-algebra on the real line. Write $|x|$ for the Euclidean norm and set

$$I=\Big(\int_{[a,b]}g^1(s)\,ds,\ \dots,\ \int_{[a,b]}g^n(s)\,ds\Big)\in\mathbb{R}^n,$$

the components being Lebesgue integrals over the compact interval $[a,b]$.

Then the map $s\mapsto|g(s)|$ is bounded and measurable, and

$$|I|\le\int_{[a,b]}|g(s)|\,ds .$$