I am reading a proof about the cádlág(right regular)-modification of a martingale. There is something in the proof I do not understand. The author defines the new process in terms of some limits, but he says that the limits can be infinite. But later in the argument he seems to treat $\tilde{X}$ as it was finite. Is this proof correct? Do you see how he excludes the case that the limits are infinity? (PS: The author only calls something a martingale if it is cádlág, if it does not have those trajectory properties it is called a martingale-structure.)
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The proof as written is incomplete, for the reason you note: for the triangle inequlity estimate to be applicable, you need to know that $\tilde X(s)$ and $\tilde X(t)$ are finite. By working a little harder, one can show that there is an $\mathcal F$-set $\Omega_1\subset \Omega_0$ such that the left and right limits of $\Bbb Q\ni r\mapsto X(r,\omega)$ exist and are finite. The additional ingredient to bring to bear (on top of the downcrossing lemma) is Doob's maximal inequality for $X$ restricted to the rationals, which shows that $\sup_{0\le r\le t, r\in\Bbb Q}|X(r)|$ is finite a.s. for each $t>0$.