I've asked myself the following simple question:
Given a right-continuous filtration $(\mathcal{F}_t)_{t \geq 0}$ on some probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and some sub-$\sigma-$algebra $\mathcal{G}\subseteq \mathcal{A}$.
Can we conclude that the "initially enlarged filtration" $(\mathcal{F}^{*}_t)_{t \geq 0}$, which is defined through $\mathcal{F}^{*}_t := \sigma(\mathcal{F}_t \cup\mathcal{G})$, is also right-continuous?
Intuitively, the r.-continuity should not vanish by adding another time-independent algebra. However, I could not translate this intuition into a rigorous proof and apparently nobody who has thought about this has written it down as a side-comment in some book or paper. At least I could not find it anywhere.
Would be greaful if you could provide an answer or share your thoughts on this with me!