I was reading paper which referred to a canonical process within the context of a measure space $(\Omega, \mathscr{F}, \mathbb{Q})$. The surrounding discussion was of functions $a:[t,T] \rightarrow \Omega$, and the canonical process was denoted $(\mathbb{S}_t)^T_{t=0}$.
It may be useful to know that the term was used when defining a "martingale measure". A measure $\mathbb{Q}$ was said to be a martingale measure if the canonical process is a local martingale with respect to the measure and $\mathbb{S}(0) = 1$ a.s.
What could this canonical process be? Any help is appreciated.
If $\{X_s:t\leq s \leq T\}$ is a given process on some probability space then, by Kolmpgorov's Theorem we can construct a measure $Q$ on $\mathbb R ^{[t,T]}$ (with the sigma algebra generated by the projection maps) such that the random variables $Y_s:t\leq s\leq T$ defined by $Y_s(\omega)=\omega (s)$ defined on $(\mathbb R^{[t,T]},Q)$ have the same finite dimensional distributions as the given process. This new process is called the canonical process corresponding to the original process. You can also start with a probability measure $Q$ on $\mathbb R^{[t,T]}$ instead of starting with a process $\{X_s:t\leq s \leq T\}$. For example if $Q$ is the Wiener measure on $C[0,1]$ then the canonical process associated with $Q$ is deined by $Y_s(\omega)=\omega (s)$ for $\omega \in C[0,1]$ and $s \in [0,1]$.