Research proof index

Convergence on a dense set of a collection of monotone function controls the limsup at a point if f is right continuous at a. We prove this under the assumption that α has both a bottom element and a top element. The bottom element is needed because otherwise limsup evaluated at the bottome element may give a junk value to break the inequality.

The constant c is integrable on the event where τₙ(c) hits before the top element.

For a submartingale indexed by a countable type, the predictable part is monotone a.e.

{τₙ(c) < 1} = {c < Aⁿ₁}.

predictableConvexStep is left-continuous in time.

The predictable part is additive for integrable processes.