Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\mathcal{N}$ be the nullsets of $\mathcal{F}$ under $P$ and $\{\mathcal{F}_t\}_{t\ge0}$ be a filtration. Is it true that $$\bigcap_{s>t} \sigma(\mathcal{F}_s\cup \mathcal{N}) = \sigma(\cap_{s>t}\mathcal{F}_s \cup \mathcal{N})$$

The inclusion from right to left is obvious but obviously the inclusion left to right is unclear to me (if true at all).


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