Null sets. Let \((X, \mathscr{A}, \mu)\) be a measure space. A set \(N \in \mathscr{A}\) is called a null set or \(\mu\)-null set if \(\mu(N)=0\). We write \(\mathscr{N}_{\mu}\) for the family of all \(\mu\)-null sets. Check that \(\mathscr{N}_{\mu}\) has the following properties:

(i) \(\emptyset \in \mathscr{N}_{\mu}\);

(ii) if \(N \in \mathscr{N}_{\mu}, M \in \mathscr{A}\) and \(M \subset N\), then \(M \in \mathscr{N}_{\mu}\);

(iii) if \(\left(N_{n}ight)_{n \in \mathbb{N}} \subset \mathscr{N}_{\mu}\), then \(\bigcup_{n \in \mathbb{N}} N_{n} \in \mathscr{N}_{\mu}\).