= Solution
The relevant fact is <uniform convergence on compacts in probability under an absolutely continuous measure change>. Write $Z=d\widetilde{\mathbb P}/d\mathbb P$. For any event $E$ and $K>0$,
$$
\widetilde{\mathbb P}(E)\leq K\mathbb P(E)+\mathbb E_{\mathbb P}[Z\mathbf1_{\{Z>K\}}].
$$
The last term tends to zero as $K\to\infty$ because $Z$ is integrable. Consequently, $\mathbb P(E_n)\to0$ implies $\widetilde{\mathbb P}(E_n)\to0$. Apply this to $E_n=\{\sup_{t\leq T}|C_t^n-[M,N]_t|>\varepsilon\}$ for each finite $T$ and $\varepsilon>0$. The given dyadic sums therefore have the same <uniform convergence on compacts in probability> limit under the new <probability measure>.
This proves <quadratic covariation under an absolutely continuous measure change>:
$$
\boxed{[M,N]^{\widetilde{\mathbb P}}=[M,N]^{\mathbb P}\quad\text{up to }\widetilde{\mathbb P}\text{-indistinguishability}.}
$$
Taking $N=M$ proves the corresponding <quadratic variation> assertion directly; it has not been assumed. The ceiling in the printed sum includes one grid interval extending beyond $t$. This does not affect the limit: on $[0,T]$ the resulting extra product is bounded by the product of the two path oscillations on mesh $2^{-n}$, which tends to zero by <uniform continuity> on $[0,T+1]$.
The bracket under the new <probability measure> must be understood as the <quadratic covariation> of continuous <semimartingales>. <Semimartingale stability under an absolutely continuous measure change> ensures this class is preserved; the <local martingale> property need not be. For example, on a finite horizon, weighting <Brownian motion> by $\exp(\mu B_T-\mu^2T/2)$ makes $B_t-\mu t$ a <Brownian motion> by the <Girsanov theorem>, so $B$ has nonzero drift under the new measure even though its <quadratic variation> is still $t$. The PDF additionally assumes that $M,N$ remain <local martingales> under the new <probability measure>, so its bracket is also defined directly by the <local martingale> characterization.
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