Semimartingale stability under an absolutely continuous measure change

If $X$ is a semimartingale under $P$ and $Q\ll P$, then it is a semimartingale under $Q$. The Bichteler-Dellacherie theorem gives a short proof: terminal elementary integrals associated with uniformly small elementary predictable processes with stopping-time intervals converge to zero in $P$-probability. Uniform convergence on compacts in probability under an absolutely continuous measure change, or its single-random-variable proof, transfers convergence to $Q$. The same good-integrator characterization now applies under $Q$. Adaptedness and càdlàg paths are preserved because $P$-null sets remain $Q$-null; if necessary use the usual $Q$-completion of the same filtration. If the integrands are bounded only outside $Q$-null sets, clip their coefficients to the specified deterministic bounds before applying the $P$-criterion; this preserves their $Q$-versions. Completion adds only equivalent versions of measurable coefficients. This theorem does not preserve the local martingale property.

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