Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / A / 4 / a / i / Solution

A Borel measurable function has $f^{-1}(B)\in \mathcal{B}$ for every Borel set $B$. Thus the upper-ray condition is necessary.

For sufficiency, note that the rays generate the stipulated sigma-algebra: $(a,b]=(a,\infty )\setminus (b,\infty )$, and $(y,\infty )={\bigcup }_{m>y}(y,m]$, with integer $m$. The familyis a sigma-algebra because inverse images preserve complements and countable unions. If it contains all upper rays, it contains their generated Borel sigma-algebra. Therefore upper-ray preimages suffice to prove Borel measurability.