さて,問題16.2を解こう. このためには, 次の $z_s \in L^1_{+1} (\Omega_s )$ を計算すればよい. \begin{align} \small{ \lim_{ \Xi_t \to x_t \;\;(t\in T)} \frac{\int_{\Omega_0} [{\widehat F}_{0}(({{{\times}}}_{t=0}^n \Xi_{t}) \times \Gamma_s)](\omega_0) \;\; z_0 (\omega_0 ) d\omega_0}{\int_{\Omega_0} [{\widehat F}_{0}({{{\times}}}_{t=0}^n \Xi_{t}) ](\omega_0)\;\; z_0 (\omega_0 ) d\omega_0} = \int_{\Omega_s} [G_s(\Gamma_s )](\omega_s) \;\; z_s (\omega_s ) d \omega_s \quad (\forall \Gamma_s \in {\mathcal F}_s ) } \end{align} 以下に, $z_s= [B_{\widehat{\mathsf{O}}_{0} }^s ({{{\times}}}_{t \in T} \{x_t \})]({z}_0)$ を計算しよう. \begin{align} & \int_{\Omega_0} [{\widehat F}_{0}(({{{\times}}}_{t=0}^n \Xi_{t}) \times \Gamma_s)](\omega_0) \;\; z_0 (\omega_0 ) d\omega_0 \nonumber \\ = & {}_{{}_{L^1(\Omega_0)}} \langle z_0, {\widehat F}_{0}(({{{\times}}}_{t=0}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_0)}} \nonumber \\ = & {}_{{}_{L^1(\Omega_1)}} \langle \Phi^{0,1}_* (F_0(\Xi_0) z_0), {\widehat F}_{1}(({{{\times}}}_{t=1}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_1)}} \tag{16.7} \end{align}
 $(A):$ ここで, $\widetilde{z}_0=F_0(\Xi_0) z_0$ (もっと正確には,その正規形 $\widetilde{z}_0=\lim_{\Xi_0 \to x_0} \frac{F_0(\Xi_0) z_0}{\int_{\Omega_0}{F_0(\Xi_0) z_0} d\omega_0}$) , $\widetilde{z}_1=F_1(\Xi_1) \Phi^{0,1}_* (\widetilde{z}_0)$, $\widetilde{z}_2=F_2(\Xi_2) \Phi^{1,2}_* ( \widetilde{z}_1)$, $\cdots$ , $\widetilde{z}_{s-1}=F_{s-1}(\Xi_{s-1}) \Phi^{s-2,s-1}_* (\widetilde{z}_{s-2})$ とおいて,さらに計算を進めて,
\begin{align} (16.7) = & {}_{{}_{L^1(\Omega_1)}} \langle \Phi^{0,1}_* (\widetilde{z}_0), {\widehat F}_{1}(({{{\times}}}_{t=1}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_1)}} \nonumber \\ = & {}_{{}_{L^1(\Omega_2)}} \langle \Phi^{1,2}_* ( \widetilde{z}_1), {\widehat F}_{2}(({{{\times}}}_{t=2}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_2)}} \nonumber \\ & \cdots \cdots \nonumber \\ = & {}_{{}_{L^1(\Omega_{s+1})}} \langle \Phi^{s,s+1}_* ( \widetilde{z}_{s}), {\widehat F}_{s+1}(({{{\times}}}_{t=s+1}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_{s+1})}} \nonumber \\ = & {}_{{}_{L^1(\Omega_{s})}} \langle \Phi^{s-1,s}_* ( \widetilde{z}_{s-1}), {\widehat F}_{s}(({{{\times}}}_{t=s}^n \Xi_{t}) \times \Gamma_s) \rangle {}_{{}_{L^\infty(\Omega_{s})}} \nonumber \\ = & {}_{{}_{L^1(\Omega_{s})}} \langle \Phi^{s-1,s}_* (\widetilde{z}_{s-1}), F_{s} (\Xi_{s})G_{s} (\Gamma_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}({{{\times}}}_{t=s+1}^n \Xi_{t}) \rangle {}_{{}_{L^\infty(\Omega_{s})}} \nonumber \\ = & {}_{{}_{L^1(\Omega_{s})}} \langle \Big( F_{s} (\Xi_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}({{{\times}}}_{t=s+1}^n \Xi_{t}) \Big) \Big( \Phi^{s-1,s}_* (\widetilde{z}_{s-1}) \Big), G_{s} (\Gamma_{s}) \rangle {}_{{}_{L^\infty(\Omega_{s})}} \tag{16.8} \end{align} となる. よって,次を得る. \begin{align} \small{ [B_{\widehat{\mathsf{O}}_{0} }^s ({{{\times}}}_{t \in T} \{x_t \})](z_0) = \lim_{ \Xi_t \to x_t \;\;(t\in T)} \frac{ \Big( F_{s} (\Xi_{s}) \Phi^{s,s+1}{\widehat F}_{s+1}({{{\times}}}_{t=s+1}^n \Xi_{t}) \Big) \times \Big( \Phi^{s-1,s}_* \widetilde{z}_{s-1}) \Big) } {\int_{\Omega_0} [{\widehat F}_{0}({{{\times}}}_{t=0}^n \Xi_{t}) ](\omega_0)\;\; z_0 (\omega_0 ) d\omega_0} } \tag{16.9} \end{align}