Let \(\left(X_{t}ight)_{t \geqslant 0}\) be a stochastic process on the measure space \((\Omega, \mathscr{A}, \mathbb{P})\) satisfying (B1)(B3). Assume further that there is some measurable set \(\Omega_{0} \subset \Omega, \mathbb{P}\left(\Omega_{0}ight)=1\), such that for all \(\omega \in \Omega_{0}\) we have \(X_{0}(\omega)=0\) and \(t \mapsto X_{t}(\omega)\) is continuous. Show that

a) \(B_{t}(\omega):=X_{t}(\omega) \mathbb{1}_{\Omega_{0}}(\omega)\) is a Brownian motion on \((\Omega, \mathscr{A}, \mathbb{P})\);

b) \(X_{t}\) is a Brownian motion on the space \(\left(\Omega_{0}, \Omega_{0} \cap \mathscr{A}, \mathbb{P}_{0}:=\mathbb{P}\left(\cdot \cap \Omega_{0}ight)ight)\).

Notation. \(\Omega_{0} \cap \mathscr{A}:=\left\{\Omega_{0} \cap A: A \in \mathscr{A}ight\}\) is the trace \(\sigma\)-algebra.