completeness of semimartingale convergence

Theorem.

Let $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{F}\mathrm{,}{\mathrm{(}{\mathrm{F}}_t\mathrm{)}}_{t\mathrm{\in }{\mathrm{R}}_{\mathrm{+}}}\mathrm{,}\mathrm{P}\mathrm{)}$ be a filtered probability space. Then, the space of semimartingales $\mathrm{S}$ forms a complete (http://planetmath.org/Complete) topological vector space under the semimartingale topology.

That is, semimartingale convergence is a vector topology (http://planetmath.org/TopologicalVectorSpace) and, for any sequence $X^n\in 𝒮$ such that $X^n-X^m\to 0$ as $m,n\to \mathrm{\infty }$ then there exists an $X\in 𝒮$ with $X^n\to X$.