Let \((X, \mathscr{A}, \mu)\) be a measure space, let \(\mathscr{F} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra and denote the collection of all \(\mu\)-null sets by \(\mathscr{N}=\{N \in \mathscr{A}: \mu(N)=0\}\). Then

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

where \(F \Delta N:=(F \backslash N) \cup(N \backslash F)\) denotes the symmetric difference, see Problem 2.2 .

Data from problem 2.2

Let \(A, B, C \subset X\). The symmetric difference of \(A\) and \(B\) is \(A \triangle B:=(A \backslash B) \cup(B \backslash A)\). Verify that

\[
(A \cup B \cup C) \backslash(A \cap B \cap C)=(A \triangle B) \cup(B \triangle C)
\]