Quadratic variation under an absolutely continuous measure change
= Quadratic variation under an absolutely continuous measure change
If $Q\ll P$ and a <continuous semimartingale> is a <semimartingale> under both measures, its quadratic variations agree $Q$-indistinguishably. The same squared-increment sums converge uniformly on compacts in probability under both measures: absolute continuity transfers the original convergence, and uniqueness of the limit identifies the two continuous versions. Equivalence of measures is unnecessary.