I hope you can help me the following problem.
Assume a Wiener space, that means a probability space $(\Omega,\mathbb F,\mathbb P)$, where $\Omega = C([0,\infty))$, $X$ is the coordinate process, $\mathbb F$ the complete filtration generated by $X$ and $\mathbb P$ a probability measure, such that $X$ is a Brownian motion.
Now let $B$ be another Brownian motion, $\mathbb Q$ the distribution of the process $\int_0^\infty 1_{\{t>1\}}\mathrm d B_t$ and $\tilde{\mathbb Q}$ the distribution of $\int_0^\infty 2\cdot 1_{\{t>1\}}\mathrm d B_t$.
Show that $\mathbb Q \sim \tilde{\mathbb Q}$ but $\mathbb Q \neq \tilde{\mathbb Q}$.
Is $X$ a local martingale under $\mathbb Q$ and $\tilde{\mathbb Q}$?
I'd really appreciate some help.