Note: This question is related to the following question about complete markets in continuous time. In the linked question, the answer mentions that complete markets in this setting is a result of the Martingale Representation theorem.
I'm trying to understand the statement of the theorem as given in its Wikipedia article:
Let $B_t$ be a Brownian motion on a standard filtered probability space $(\Omega, \mathcal F, \mathcal F_t, P)$, and let $\mathcal G_t$ be the augmentation of the filtration generated by $B$. If $X$ is a square integrable random variable measurable with respect to $\mathcal G_\infty$ then there exists a predictable process $C$ which is adapted with respect to $\mathcal G_t$, such that $$ X = E[X] + \int_0^\infty C_s dB_s. $$ Consequently, $$ E[X \mid G_t] = E[X] + \int_0^t C_s dB_s. $$
In this definition, where is the relationship to Martingales? I see that it is assumed that $X$ is square integrable with respect to $\mathcal G_\infty$ and I assume that it has something to do with the fact that $\mathcal G_t$ is an "augmentation of the filtration generated by $b$." Also, what is an "augmentation of a filtration?"