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

Use the characterization of measurability by preimages of open half-lines, allowing intermediate extended-real-valued functions. Put $u_m={\inf }_{n\ge m}{f}_n$. For each real $a$,Thus every $u_m$ is measurable. Countable suprema are measurable too, since $\{{\sup }_m{u}_m>a\}={\bigcup }_m\{u_m>a\}$ and the upper half-lines also generate the Borel sigma-algebra. Thereforeis measurable. Its values are real by the stated hypothesis, so it is the required ordinary real-valued measurable function, rather than only an extended-real-valued one.