Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / A / 1 / a / i

A map between the measurable spaces is measurable whenA real-valued function $f$ on $S$ is $\Sigma$-measurable when it is measurable from $(S,\Sigma )$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Equivalently each set $\{f<a\}$ is in $\Sigma$, since open half-lines generate the Borel sigma-algebra.

A Borel measurable function on the real line is a function measurable with respect to the Borel sigma-algebra on both its domain and range. Continuity is sufficient for this property, but is not required.