Consider one-dimensional Lebesgue measure \(\lambda\) on \(([0,1], \mathscr{B}[0,1])\). Compare the convergence behaviour (a.e., \(\mathcal{L}^{p}\), in measure) of the following sequences:

(i) \(f_{i, n}:=n \mathbb{1}_{[(i-1) / n, i / n]}, n \in \mathbb{N}, 1 \leqslant i \leqslant n\), run through in lexicographical order;

(ii) \(g_{n}:=n \mathbb{1}_{(0,1 / n)}, n \in \mathbb{N}\);

(iii) \(h_{n}:=a_{n}(1-n x)^{+}, n \in \mathbb{N}, x \in[0,1]\) and a sequence \(\left(a_{n}ight)_{n \in \mathbb{N}} \subset \mathbb{R}^{+}\).