Let \((\Omega, \mathscr{A})\) be a measurable space and \(\xi: \mathbb{R} \times \Omega ightarrow \mathbb{R}\) be a map such that \(\omega \mapsto \xi(t, \omega)\) is \(\mathscr{A} / \mathscr{B}(\mathbb{R})\)-measurable and \(t \mapsto \xi(t, \omega)\) is left- (or right-)continuous. Show that the following functions are measurable:

\[t \mapsto \xi(t, \omega) \quad \text { and } \quad \omega \mapsto \sup _{t \in \mathbb{R}} \xi(t, \omega)\]

[approximate \(t \mapsto \xi(t, \omega)\) by simple functions.]