**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:

Thus:

By monotone convergence

By Doob’s Inequality for discrete martingales

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

Then

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