Completeness of continuous square-integrable martingales ($\Vert M{\Vert }_{{\mathcal{M}}^2[0,T]}=(\mathbb{E}|M_T{|}^2{)}^{1/2}$)

Continuous square-integrable martingales on a fixed finite horizon form a complete space under the displayed terminal norm, modulo indistinguishability of stochastic processes. A terminal-norm Cauchy sequence admits a subsequence whose successive differences have norms below $2^{-n}$. The Doob L2 maximal inequality and Markov's inequality make the probabilities that these differences have suprema above $2^{-n/2}$ summable. The Borel-Cantelli first lemma then gives an almost surely uniformly convergent telescoping series, so the limit has continuous paths.

Conditional-expectation contractivity gives convergence in $L^2$ at every fixed time, and passage to the limit in the martingale identities proves that the limit is a martingale. Adaptedness can be retained by taking the pointwise finite subsequence limit when it exists, and zero otherwise. The original Cauchy sequence converges in terminal norm to this limit. This completeness allows the Itô isometry to extend stochastic integration from simple predictable integrands to the full square-integrable integrand space while retaining a continuous martingale version.