Right-continuous local martingales

Question

I‘m currently studying local martingales and I am working with the following Proposition:

I do not understand why we need right-continuity for the proof. Can somebody help?

Answer 1

In general $\sup_{t \in \mathbb{R}_+} |X_t|$ need not be measurable since you take the supremum over an uncountable set. Without an assumption giving measurability, the quantity assumed to be finite in the proposition doesn't make sense. Thankfully, you can use right-continuity to say that the supremum is the same as $\sup_{t \in \mathbb{Q}_+} |X_t|$ and hence is measurable.