The question comes from the proof of Theorem 4.15 of this book.
Let $f:[0,T]\to \mathbb{R}^n$ be a bounded Lebesgue integrable function. Then, \begin{equation} \lim_{N\to\infty} \min_{(x_1,\dots,x_N)\in\mathbb{R}^{nN}} \sum_{i=1}^N \int_{(i-1)T/N}^{iT/N}\|f(t) - x_i\|dt = 0 \end{equation} where the norm is Euclidean.
Would you give me any hint or a reference?
Hint: Note that
$$\sum_{i=1}^N\int_{(i-1)T/N}^{iT/N}\Vert f(t)-x_i\Vert dt=\Vert f-s\Vert_1$$
where $\Vert\cdot\Vert_1$ is the $L^1$-norm, and $s:[0,T]\to\mathbb{R}^n$ is the stair function with $s(x)=x_i$ if $(i-1)T/N\leq x In other words, the problem is to prove that such stair functions are dense in $L^1([0,T],\mathbb{R}^n)$. You can prove this by approximating function in $L^1$ by continuous functions, then using uniform continuity to find a stair function as above which approximates the continuous function. Note that this approach doesn't require boundedness at any point.