= 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.
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