Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / A / 1 / b

The intersection of two nonempty members iswhich belongs to the collection. Any intersection involving the empty set is empty and is included too. Hence $\pi (\mathbb{R})$ is a pi-system.

Each half-line $(-\infty ,x]$ is a closed Borel set, and the empty set is Borel. Thus $\sigma (\pi (\mathbb{R}))\subseteq \mathcal{B}(\mathbb{R})$.

Conversely,The generated sigma-algebra contains both sets on the right and is closed under their difference, so it contains every open interval. By the given definition of the Borel sigma-algebra, the reverse inclusion follows. Therefore