In the probability space $(\Omega,\mathcal{A},\mathbb{P})$ let $\mathcal{F} = \{ \mathcal{F}_t : t\in\mathbb{R} \}$ be a Filtration, $\mathcal{A}_0 \subset \mathcal{A}$ a sub-$\sigma$-field and $\tau$ a $\mathcal{F}$-stopping time. Define the enlarged filtration $\mathcal{G} = \{ \mathcal{G}_t : t\in\mathbb{R}\}$ through $\mathcal{G}_t = \sigma(\mathcal{F}_t \cup \mathcal{A}_0)$; $\tau$ is also a $\mathcal{G}$-stopping time. The stopped $\sigma$-field $\mathcal{F}_\tau$ is defined as the collection of all sets $A\in\mathcal{A}$ with $A\cap\{\tau\leq t\}\in\mathcal{F}_t$ for all $t\in\mathbb{R}$; and $\mathcal{G}_\tau$ similar.
Is $\mathcal{G}_\tau = \sigma(\mathcal{F}_\tau \cup \mathcal{A}_0)$ true? I'm especially interested in $\mathcal{G}_\tau \subset \sigma(\mathcal{F}_\tau \cup \mathcal{A}_0)$.
Thx for any help in advance!