Let $(\Omega, \mathcal{F},\mathbb{F}, P)$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}_t,\,t\ge 0\right)$ is a filtration. I am curious about the rigorous definition of $\mathcal{F}_{t-}$ for $t>0$ (which I often see used in the context of pure jump processes). Although the intuitive description of $\mathcal{F}_{t-}$ is clear, I could not find a book reference or a suitable post on Mathematics Stack Exchange that provides a proper mathematical definition. My guess is that $$\mathcal{F}_{t-} := \sigma\left(\bigcup_{0\le s<t} \mathcal{F}_s\right).$$ Is this correct?
On a similar note, suppose we have a $P$-a.s. left-continuous stochastic process $X=\left(X_t,\,t\ge 0\right)$ defined on this space and let $\mathbb{F}^X = \left(\mathcal{F}^X_t,\,t\ge 0\right)$ be the filtration induced by $X$, i.e., $\mathcal{F}_t:=\sigma\left(X_s^{-1}(B),\,0\le s\le t,\, B\in\mathcal{B}_{\mathbb{R}}\right)$. Then $X_{t-}:=\lim_{s\uparrow t} X_s$ exists almost surely for each $t>0$. Are the following two equalities then true (?): $$\mathcal{F}_{t-}^X = \sigma\left(X_s^{-1}(B),\,0\le s<t,\, B\in\mathcal{B}_{\mathbb{R}}\right) = \sigma\left(\mathcal{F}^X_0\cup\left\{X_{s-}^{-1}(B),\,0<s<t,\, B\in\mathcal{B}_{\mathbb{R}}\right\}\right).$$ (In the second equality above I assume we need to add $\mathcal{F}^X_0$ to capture the information of the process $X$ at time $t=0$.) Finally, I assume that in general we would not have $\mathcal{F}_{t-}=\mathcal{F}_t$ for all $t>0$, correct (e.g., if $X$ is a pure jump process)?