Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 29 / 2 / b / Solution

Fix a sample path on the probability-one event where $X$ is continuous and the total-variation process of $A$ satisfies $V_{\infty }\le K$. The signed measure $dA$ has absolute variation measure $dV$. ThusThe integrand tends to zero at every time, including times when $X_s=0$, and is bounded by $2$. Since $dV$ is a finite measure on this path, the dominated convergence theorem makes the right-hand side tend to zero. ThereforeIn fact the convergence is uniform over the entire time axis under the given total-variation bound. This pathwise estimate establishes one common full-probability event, rather than separate exceptional events for each time.