# Filtration related to stopping time of Brownian Motion: Paths are measurable?

Given a stopping time $$T:\Omega\rightarrow \mathbb{R}$$ there is a related $$\sigma$$-algebra used in Mörters and Peres book, Brownian Motion defined on p. 42 as $$\mathcal{F}^+(T) = \{A\in \mathcal{A}: A\cap \{T\leq t\}\in \mathcal{F}^+(t)\text{ for all t\geq 0}\}$$ where $$\mathcal{A}$$ is a $$\sigma$$-algebra and $$\mathcal{F}^+(t)$$ is a filtration related to Brownian motion via $$\mathcal{F}^+(t) = \bigcap_{s>t}\sigma(B(\tau):0\leq \tau\leq s).$$ Now on p. 43 they claim that $$\{B(t): t\leq T\}$$ is $$\mathcal{F}^+(T)$$ measurable. However I'm confused as to what this means. $$\mathcal{F}^+(T)$$ is clearly a $$\sigma$$-algebra on $$\Omega$$ while $$\{B(t):t\leq T\}$$ is for each $$\omega\in \Omega$$ a subset of $$\mathbb{R}$$? Could someone clarify this for me?

We have for example \begin{align*} \sigma ( B(\tau) : 0 \leq \tau \leq s) &= \sigma (\lbrace B(\tau) \in B \rbrace: 0\leq \tau \leq s, B\in\mathcal{B}(\mathbb{R})) \\ &= \sigma (B(\tau)^{-1}(B):0\leq \tau \leq s, B \in \mathcal{B}(\mathbb{R})) \end{align*} and now it is more clear that the $$\sigma$$-algebra is actually generated by sets in $$\Omega$$.
Regarding the set $$\lbrace B(t):t\leq T\rbrace$$ it seems to me that this is defined on the path space in the same way that the process $$\lbrace B(t):t\geq 0\rbrace$$ would be. So $$B(\cdot,\omega)$$ is considered a mapping from $$\Omega$$ to the space of continuous functions $$C([0,\infty),\mathbb{R})$$.