Let $\left\{ X_{t}\right\} _{t\geq o}$ the canonical version of Brownian motion, i.e., if we consider $\Omega:=\mathbb{R}^{\left[0,\infty\right)}$ the set of the real valued functions on $\left[0,\infty\right)$ and for each $t\geq0$, $X_{t}\left(\omega\right)=\omega\left(t\right)$ and $\mathcal{F}:=\sigma\left\{ \omega\in\Omega|X_{t_{1}}\left(\omega\right)\in B_{1},\ldots,X_{t_{k}}\left(\omega\right)\in B_{k},B_{i}\in\mathcal{B}\left(\mathbb{R}\right)\right\} $ then the process $\left\{ X_{t}\right\} _{t\geq o}$ is called the canonical version of Brownian motion. Proof that
$$A=\left\{ \omega\in\Omega|\sup_{a\leq t\leq b}X_{t}\left(\omega\right) for some $c\in\mathbb{R}$. I cheked Introduction to Stochastic Integration-Kuo Hui-Hsiung, but
here there is a particular case. However, I think I have to find a
nonmeasurable set on $\mathcal{F}$
and then proceed in the similar way, but I have been unable to make
any headway in that respect.