Doob’s Inequality for Right (Left) Continuous Martingales
In all the proofs I have seen, some of the details are elided; here is my proof with all the details.
Proof: Let be an increasing set of finite sets whose union is dense in and note that by right (left) continuity, we have:
By monotone convergence
Note that if is not continuous then all bets are off. For example, consider the finite sets on the interval of dyadic rationals:
Then is dense in and consider the function