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For a continuous semimartingale $X$, its quadratic variation is the continuous increasing zero-starting process obtained as the limit, uniformly on compact time intervals in probability, of sums of squared increments along deterministic partitions whose mesh tends to zero. If $X=X_0+N+V$ is its continuous local-martingale/finite-variation decomposition, then $[X]=[N]$; finite-variation terms and their cross sums vanish.

Write $D=d\mathbb{Q}/d\mathbb{P}$. If $\mathbb{P}(E_n)\to 0$, then for every $R>0$,First send $n$ to infinity and then $R$ to infinity. Thus convergence in probability under $\mathbb{P}$ implies convergence in probability under $\mathbb{Q}$, also for a supremum on a compact time interval. The same squared-increment sums converge under $\mathbb{Q}$ to the $\mathbb{P}$ version of $[X]$. Since $X$ is also a semimartingale under $\mathbb{Q}$, these sums converge to its $\mathbb{Q}$ quadratic variation. Uniqueness of limits in probability gives equality at every fixed time; continuity and countably many rational times giveAbsolute continuity suffices; equivalence is not required. This is quadratic variation under an absolutely continuous measure change.