Let (left(B_{t}, mathscr{F}_{t} ight)_{t geqslant 0}) be a (mathrm{BM}^{1}) such that (mathscr{F}_{0}) contains all measurable null sets

Let \(\left(B_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) such that \(\mathscr{F}_{0}\) contains all measurable null sets and let \(\tau\) be a stopping time. Show that \(B_{\tau}\) is measurable with respect to \(\mathscr{F}_{\tau+}\).

Use in Theorem \(6.6 u(x)=u_{n}(x) \rightarrow x, u \in \mathcal{C}_{b}\left(\mathbb{R}^{d}\right)\); see also Corollary 6.26.

Data From Corollary 6.26

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6.26 Corollary. Let (B+)to be a BMd and be an F+ stopping time. Then B+17 <00} is F+ measurable. Proof. Consider the maps Ex. 6.7 w(T(w),w) (s,w)B(s,w) ({T t}, Ft) ([0, t], [0, t] Ft) (Rd, B(Rd)). The second map is measurable, since (Bt)zo is progressively measurable. For the first map we observe that for all r