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Work modulo indistinguishability of stochastic processes, with norm $\Vert M\Vert =(\mathbb{E}|M_t{|}^2{)}^{1/2}$. It is a genuine norm: zero terminal value implies $M_s=\mathbb{E}[M_t\mid {\mathcal{F}}_s]=0$ at every fixed time, and continuity extends equality from countably many rational times to all times simultaneously.

Let $M^j$ be a Cauchy sequence in this norm. Choose a subsequence with $\Vert M^{j_{n+1}}-M^{j_n}\Vert <2^{-n}$. Part (b) applies to these difference martingales, so their telescoping series converges uniformly almost surely. Its limit has continuous paths. At each time $s$, conditional-expectation contractivity givesso the fixed-time limit also exists in $L^2$ and agrees with the pathwise limit. It has an adapted version obtained by taking the finite pointwise subsequence limit when it exists, and zero otherwise. Each such definition is measurable at its own time, and the almost surely uniform limit gives continuous paths. Passing to the $L^2$ limit in $\mathbb{E}[M_v^j\mid {\mathcal{F}}_s]=M_s^j$ shows that the limit is a martingale. The original Cauchy sequence converges to it in terminal norm. Thus the space is complete.

Part (b) supplies what fixed-time $L^2$ completeness alone does not: a single full-probability set on which a subsequence converges uniformly in time, preserving path continuity. This is completeness of continuous square-integrable martingales.

For continuous semimartingales and $Q\ll P$, their quadratic covariation has the same version under the two probability measures, up to $Q$-indistinguishability of stochastic processes. The dyadic product sums have the same limit by uniform convergence on compacts in probability under an absolutely continuous measure change. Uniqueness of a limit in probability identifies that limit with the bracket under $Q$. Semimartingale stability under an absolutely continuous measure change supplies the correct class under $Q$: a local martingale need not remain a local martingale. Taking the two processes equal proves the same statement for quadratic variation without a separate assumption.

Two Ornstein-Uhlenbeck processes with the same driving Brownian motion have the displayed difference. If $Y_0$ is standard normal and independent of future Brownian increments, replacing it by $-Y_0$ produces another stationary process with the same covariance and second moment, but a different pathwise coupling. Equal initial values are necessary for indistinguishability of stochastic processes.