Let \((X, \mathscr{A})\) be a measurable space.

(i) Let \(f, g: X ightarrow \mathbb{R}\) be measurable functions. Show that for every \(A \in \mathscr{A}\) the function \(h(x):=f(x)\), if \(x \in A\), and \(h(x):=g(x)\), if \(x otin A\), is measurable.

(ii) Let \(\left(f_{n}ight)_{n \in \mathbb{N}}\) be a sequence of measurable functions and let \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) such that \(\bigcup_{n \in \mathbb{N}} A_{n}=X\). Suppose that \(\left.f_{n}ight|_{A_{n} \cap A_{k}}=\left.f_{k}ight|_{A_{n} \cap A_{k}}\) for all \(k, n \in \mathbb{N}\) and set \(f(x):=f_{n}(x)\) if \(x \in A_{n}\). Show that \(f: X ightarrow \mathbb{R}\) is measurable.