Stochastic Calculus: I

The \(\sigma \) -algebra generated by a random variable X, denoted \(\sigma (X)\) , is the \(\sigma \) -algebra by the collection of \(\{X \in B\}\) (i.e., \(\{\omega : X(\omega ) \in B\}\) ), where B is any interval in \({\mathbb {R}}\) . Let \(\mathscr {G}\) be a \(\sigma \) -algebra on \(\varOmega \) . Then X is said to be \(\mathscr {G}\) -measurableMeasurable if \(\sigma (X) \subseteq \mathscr {G}\) .