 Stochastic integral under an absolutely continuous measure change

If $Q\ll P$ and $X$ is a semimartingale under both measures, the two integrals of any locally bounded predictable $H$ against $X$ agree up to $Q$-indistinguishability. Elementary integrals are the same increment sums. Bounded pointwise convergence of integrands, stochastic dominated convergence, and transfer of probability convergence from $P$ to $Q$ extend equality by the monotone-class theorem. Localize to remove boundedness.

// Target: probability-and-statistics.bigb