Let $(\Omega,\mathcal A)$ be a probability space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $\tau:\Omega\to[0,\infty]$. Unfortunately, there is no common definition of the "$\sigma$-algebra of the strict $\tau$-past" $\mathcal F_{\tau-}$. In the literature, you will find \begin{align}\mathcal F^1_{\tau-}&:=\sigma\left(\bigcup_{t\ge0}\left.\mathcal F_t\right|_{\{\:t\:<\:\tau\:\}}\right)=\sigma\left(\left\{A\cap\left\{t<\tau\right\}:A\in\mathcal F_t\text{ and }t\ge0\right\}\right)\\\mathcal F^2_{\tau-}&:=\sigma\left(\mathcal F_0\cup\left\{A\cap\left\{t<\tau\right\}:A\in\mathcal F_t\text{ and }t>0\right\}\right)\\\mathcal F^3_{\tau-}&:=\sigma\left(\mathcal F_{0+}\left\{A\cap\left\{t<\tau\right\}:A\in\mathcal F_t\text{ and }t\ge0\right\}\right).\end{align} Note that we usually define $$\mathcal F_{t-}:=\sigma\left(\bigcup_{s\in[0,\:t)}\mathcal F_s\right)$$ for $t\in(0,\infty)$ and I think a sensible definition of $\mathcal F_{\tau}$ should be equal to $\mathcal F_{t-}$ whenver $\tau\equiv t$ for some $t\in(0,\infty)$.
So, my queston is: Is any of the definitions above superior? Are there any advantages/disadvantages of them?