Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\mathbb{T}$ be an ordered index set (usually, the set of time periods). I encountered a filtration $(\mathcal{F}_{t})_{t \in \mathbb{T}}$ defined as:
- The filtration $(\mathcal{F}_{t})_{t \in \mathbb{T}}$ is generated by the coordinate functions.
Although I have checked several books as well as searched several online documents, I could not find the definition of "generated by the coordinate functions."
I'd appreciate it if you would provide the definition or some references. Thank you.
I do not know whether it corresponds to your setting. However, there are cases where we can define the filtration generated by the coordinate functions. Assume that $\Omega$ is a product space, say $\Omega=\prod_{t\in\mathbb T}\Omega_t$, where $\left(\Omega_t,\mathcal A_t,\Pr_t\right)$ are probability spaces. Then let $\mathcal F_t$ be the $\sigma$-algebra generated by the $\mathcal A_{t'}$, $t'\leqslant t$ for the order on $\mathbb T$. This is the smallest $\sigma$-algebra making the projections $\pi_{t'}$ on $\Omega_{t'}$ measurable for all $t'\leqslant t$, hence the name.