In our analysis course, we just defined the following:
Let $g := (g_1, \ldots, g_n): [a, b] \to \mathbb{R}^n$, where $g_1, \ldots, g_n: [a,b] \to \mathbb{R}$ are integrable. Then we call the integral of $g$ over $[a,b]$ \begin{equation*} \int_{a}^{b} g(t) \ dt := \begin{pmatrix} \int_{a}^{b} g_1(t) \ dt \\ \vdots \\ \int_{a}^{b} g_n(t) \ dt \end{pmatrix}. \end{equation*}
I came across this definition again at the beginning of measure theory, when we stated:
We ultimately want to integration functions $\mathbb{R}^n \to \mathbb{R}^m$, but because of the above definition we can, without loss of generality, restrict ourselves to the case $m = 1$.
My Question What is the intuition behind this definition, why does it make sense, if you will, ''on a deeper level''