Null sets are sometimes important! Let S and F be as in Show that where is a continuous function on [0, 1]}. Suppose X is a stochastic process, its minimal augmented filtration, and Suppose...

Null sets are sometimes important! Let S and F be as in Show that

where

is a continuous function on [0, 1]}.

Suppose X is a stochastic process,

its minimal augmented filtration, and

Suppose with probability one, the paths of X are right continuous with left limits. Let

the left-hand limit at time t, and

the size of the jump at time t. If

Prove