Let $X:\Omega\times[0,T]\to\mathbb R$ be a continuous local martingale on a complate and right-continuous filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$. I want to show that $$\int_0^\cdot\frac{\left|X_{s+\varepsilon}-X_s\right|^2}\varepsilon\:{\rm d}s\to[X]\tag1$$ uniformly in probability. How can we do that?
I've got some loose ideas: If $X$ is square-integrable, then $$\operatorname E\left[\int_0^t\frac{\left|X_{s+\varepsilon}-X_s\right|^2}\varepsilon\:{\rm d}s\right]=\int_0^t\operatorname E\left[\frac{\left|X_{s+\varepsilon}-X_s\right|^2}\varepsilon\right]\:{\rm d}s=\int_0^t\operatorname E\left[\frac{[X]_{s+\varepsilon}-[X]_s}\varepsilon\right]\:{\rm d}s$$ by Fubini (and since $\operatorname E[|X_t|^2]=\operatorname E[[X]_t]$) for all $t\in[0,T]$. So, maybe we've got $L^2$-convergence for the product measure. Then, this result could be used for a localizing sequence.