Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / A / 1 / c / iii

For each $i$, the collection of events $\{X_i\le x\}$, together with the empty event, is a pi-system. Its generated sigma-algebra is $\sigma (X_i)$, because the half-lines generate the Borel sigma-algebra and inverse images preserve intersections and the sigma-algebra operations.

The displayed probability identity says precisely that these two pi-systems are independent. Part (ii) makes $\sigma (X_1)$ and $\sigma (X_2)$ independent, which is the definition of independent random variables. Conversely independence of the random variables applies to the half-line events and gives the identity. Thus the joint cumulative distribution function factors exactly when the variables are independent.