= Completeness of continuous square-integrable martingales
{title2=$\|M\|_{\mathcal M^2[0,T]}=(\mathbb E|M_T|^2)^{1/2}$}
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.
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