Let $(\Omega^1, \mathcal{F}^1)$ and $(\Omega^2,\mathcal{F}^2)$ be two measurable space and let $(\mathcal{F}^2_s)_{s \geq 0}$ be a filtration on $(\Omega^2,\mathcal{F}^2)$. Moreover, let $t\geq 0$ be fixed.
My question is the following: Does the following equation hold true?:
$\mathcal{F}^1 \otimes \bigcap_{\varepsilon>0} \mathcal{F}^2_{t+ \varepsilon} = \bigcap_{\varepsilon>0} \big[ \mathcal{F}^1 \otimes \mathcal{F}^2_{t+ \varepsilon} \big]$