When reading the book Measures, Integrals and Martingales written by R.L. Schilling, I saw a statement as below: $\mathcal{A} \times \mathcal{B}:= \{A \times B: A\in \mathcal{A}, B\in \mathcal{B}\}$ where $\mathcal{A}$ and $\mathcal{B}$ are $\sigma$-algebras, is not $\sigma$-algebra in general.
However, I cannot construct a counterexample. Could anyone offer help here?
Kind regards