Let \(\overline{\mathscr{A}}\) denote the completion of \(\mathscr{A}\) as in Problem 4.15 and write

\[\mathscr{N}:=\{N \subset X: \exists M \in \mathscr{A}, N \subset M, \mu(M)=0\}\]

for the family of all subsets of \(\mathscr{A}\)-measurable null sets. Show that

\[\overline{\mathscr{A}}=\sigma(\mathscr{A}, \mathscr{N})=\{A \Delta N: A \in \mathscr{A}, N \in \mathscr{N}\}\]

Conclude that for every set \(A^{*} \in \overline{\mathscr{A}}\) there is some \(A \in \mathscr{A}\) such that \(A \triangle A^{*} \in \mathscr{N}\).

Data from problem 4.15

 Completion (1) We have seen in Problem 4.12 that measurable subsets of null sets are again null sets: \(M \in \mathscr{A}, M \subset N \in \mathscr{A}, \mu(N)=0\) then \(\mu(M)=0\); but there might be subsets of \(N\) which are not in \(\mathscr{A}\). This motivates the following definition: a measure space \((X, \overline{\mathscr{A}}, \mu)\) (or a measure \(\mu\) ) is complete if all subsets of \(\mu\)-null sets are again in \(\overline{\mathscr{A}}\). In other words, it holds if all subsets of a null set are null sets.
The following exercise shows that a measure space \((X, \mathscr{A}, \mu)\) which is not yet complete can be completed.