Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(A_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{A}\) be mutually disjoint sets such that \(X=\biguplus_{n \in \mathbb{N}} A_{n}\). Set

\[Y_{n}:=\left\{u \in L^{2}(\mu): \int_{A_{n}^{c}}|u|^{2} d \mu=0ight\}, \quad n \in \mathbb{N}\]

(i) Show that \(Y_{n} \perp Y_{k}\) if \(n eq k\).

(ii) Show that \(\operatorname{span}\left(\biguplus_{n \in \mathbb{N}} Y_{n}ight)\) (i.e. the set of all linear combinations of finitely many elements from \(\left.\biguplus_{n \in \mathbb{N}} Y_{n}ight)\) is dense in \(L^{2}(\mu)\).

(iii) Find the projection \(P_{n}: L^{2}(\mu) ightarrow Y_{n}\).