Doing some exercises from a mathematical finance book, I got stuck at the following point. It is a purely probability question. Let $S_t^1 = \sigma W_t$, where $W_t$ is a brownian motion and $\sigma>0$ a parameter. Furthermore let $K>0$ also be a positive constant. I want to compute the price of a call option under $Q$, i.e.
$E_Q[(S_T^1-K)^+|\mathcal{F}_t]$
So far I was able to do this: Let $A:=\{S^1_T>K\}$
$E_Q[(S_T^1-K)^+|\mathcal{F}_t]=E_Q[S_T^1\mathbf1_A|\mathcal{F}_t]-KE_Q[\mathbf1_A|\mathcal{F}_t]$
Writing $S^1_T=S_t^1+\sigma(W_T-W_t)$ leads to
$\sigma E_Q[(W_T-W_t)\mathbf1_A|\mathcal{F}_t]+(S^1_t-K)E_Q[\mathbf1_A|\mathcal{F}_t]$
Now here is the point, where I got stuck. I know $(W_T-W_t)$ is independent of $\mathcal{F}_t$ but I do not see if $A\in \mathcal{F}_t$. Or how else should I simplify this?