Characterization of a set in the augmented filtration $\mathcal{F}_t^{\mu}=\sigma(\mathcal{F}_t^X, \mathcal{N}^{\mu})$

Question

Can I write any set in $\mathcal{F}_t^{\mu}=\sigma(\mathcal{F}_t^X, \mathcal{N}^{\mu})$ as $A\cup B$ where $A \in \mathcal{F}_t^X $ and $B \in \mathcal{N}^{\mu}$ ?

I am quite convinced that this is true due to the following theorem below

Am I correct?