$(X_t,\mathcal{F}_t)$ is called a Weiner martignale if i) $X_t$ is a Wiener Process ii) $(X_t,\mathcal{F}_t)$ is a martingale. (Here $\mathcal{F}_t$ is an increasing $\sigma$-field family).
Let $(\Omega,\Sigma,P)$ be a complete probability space on which we have Wiener process, $(W_t)$. Denote $\mathcal{F}_t^W = \mathcal{F}_t^{W,0} \bigvee \{P-\mbox{null sets of }\Omega\}$ where $\mathcal{F}_t^{W,0} = \sigma(W_s, s\leq t)$.
It can be shown that $(W_t,\mathcal{F}_t^W)$ is a Wiener Martingale. The book I am reading says that it is easy to find $\mathcal{F}_t \neq \mathcal{F}_t^W$ such that $(W_t,\mathcal{F}_t)$ is also a Wiener Martingale.
I think that $\mathcal{F}_t \supset \mathcal{F}_t^{W,0}$ as otherwise $E[W_t | \mathcal{F}_t]$ might not be equal to $W_t$. On the other hand, $E[W_s - W_t | \mathcal{F}_t]$ ($s > t$) might not be zero if $\mathcal{F}_t \supset \mathcal{F}_t^{W,0}$. So I am not sure how to choose $\mathcal{F}_t$ such that $(W_t,\mathcal{F}_t)$ is also a martingale and $\mathcal{F}_t \neq \mathcal{F}_t^W$. ($\mathcal{F}_t = \mathcal{F}_t^{W,0}$ is one such choice but I want to know if any others exist)
Hint: Let $X$ be a (non-trivial) random variable such that $X$ is independent from $\mathcal{F}^W_{\infty} := \sigma(\mathcal{F}_t^W; t \geq 0)$. Consider the filtration defined by $$\mathcal{F}_t := \sigma(\mathcal{F}_t^W, X).$$