Let \(\Omega\) be a set and \(B, B^{c} \subset \Omega\) such that \(B\) and \(B^{c}\) are not empty.

(i) Find all measurable functions \(u:(\Omega,\{\emptyset, \Omega\}) ightarrow(\mathbb{R}, \mathscr{A})\), if (a) \(\mathscr{A}:=\{\emptyset, \mathbb{R}\}\), (b) \(\mathscr{A}:=\) \(\mathscr{B}(\mathbb{R})\) and (c) \(\mathscr{A}:=\mathscr{P}(\mathbb{R})\).

(ii) What does \(u \in \mathcal{L}^{p}(\Omega, \sigma(B), \mu), p>0\), look like?